Daniel Dockery

animî nostrî dêbent interdum âlûcinâri

Home of published musician, recording artist, mathematician, programmer, translator, artist, classicist, and general polymath.

Polygorials

June 21st, 2003

A little while ago I ran across and couldn’t resist the urge to acquire a copy of the last published critical edition of Nicomachus of Gerasa‘s nearly 2000-year-old Introduction to Arithmetic (Αριθμητικη εισαγωγη), the Hoche edition published by Teubner in Leipzig in 1866.

Cover of 1866 Introduction to Arithmetic

The provenance of the volume is interesting in itself. A small imprint on the inside front cover, “Paul Koehler—Buchhändler und Antiquar—Leipzig ’05“, shows where, thirty-nine years after its publication, it went through the bookshop of the famed German antiquarian and publisher Paul Koehler; in the rear, on the blank bottom of the last page of the index, is a stamp in Hebrew, stating that it was at one point part of the library collection of the University of Jerusalem; then on the FEP, the handwritten signature of the late, renowned philologist, linguist and classical scholar, Benedict Einarson (long of U. Chicago’s classics dept.). When I found the volume, it had already made its way back to a small bookshop in the UK.

In a recent period of recovery, I kept myself occupied by reading the little volume, and while so engaged found myself entertaining myriad ideas about polygonal numbers, a dormant fascination reawakened by Nicomachus. Skipping a good bit of middle ground and jumping to the point: I generalized the idea of the factorial function, which traditionally produces only the product of the first n natural or counting numbers, to a function which returns the product of the first n polygonal numbers for k-sided polygons. For simplicity of reference, I have called these “polygorials”, merging the two words polygonal factorials. In exploring these I discovered that a few have already appeared in the literature as other sequences in the Encyclopedia of Integer Sequences, though most were novel to this research, which also yielded other surprises, such as the fact that the ratio of the nth hexagorial to the nth trigorial yields the Catalan numbers.

For those interested, a small, introductory paper on the Polygorials details the work.


Bulgarian Goats

May 15th, 2003

Before realizing it was a current problem in the 2003, Term 2, “Mathematics Challenge for Young Australians”, some mailing list comments on the “Bulgarian Goats” problem elicited some thought, code and diagrams which I’ll share here.

The problem:

Yenko the Bulgarian goatherd drives his father’s goats into a valley each morning and lets them browse there all day before driving them home in the evening.

He notices that each morning the goats immediately separate into groups and begin to feed. The number and sizes of the initial groups vary. Some days there are nine or more groups; on other days, there are three or fewer. There can be groups of one or the whole herd can form a single group.

About every five minutes one goat breaks away from each feeding group and these breakaway goats form into a new group.

Yenko has noticed that by the afternoon, even though the goats continue their regrouping, the sizes of the groups have stabilized, and there are always seven feeding groups.

a.) How many goats are there in the herd?

What are the sizes of the feeding groups once they have stabilized?

Yenko’s father then sells two of the goats. Over the next week, Yenko notices that things have changed. The sizes of the feeding groups no longer stabilize. There are not always seven groups. Nevertheless, a cyclic pattern of sizes develops every day.

b.) Find at least two possible cyclic patterns of sizes.

If we say n is the total number of goats, then the number of possible group configurations into which they can separate is p(n)—where p(n) represents the number of integer partitions of n—since any division of a herd with n members into groups is merely a partition of n. E.g., if there are 3 goats there can be anywhere from one to three groups corresponding to the three partitions of 3: {3}, {2,1} and {1,1,1}. If there is one group of 3, and one goat leaves to form a new group, {3} becomes {2,1}; if, again, one goat leaves from each group and they join to form a new one, {2,1} recurs. If they begin in three groups of 1, and leave to form a new group the cycle goes from {1,1,1} to {3} to {2,1} which latter repeats as before.

It seems to me that for a herd partition to reproduce itself, it must have a number of groups equal to the size of the largest group. Since at each iteration we add a new group [with a number of members equal to the total number of groups (the new group being composed of the goats which separated from each of the previous groups)], in order to keep the same number of values—which we must do, obviously, if we’re going to reproduce the same partition—we must also lose a group, and since we subtract one from each group at each iteration the way to do that is for us to have one group with only 1 member. Since we must have a group with 1 member, however, we must also have a group with 2 so that 1 will recur when one member of each group separates to form the new group. And it’s easy to see that for each number x we require to recur—with m, the new group, being, as before, the largest value—we must also have an x+1; since our smallest value must be 1, this means we must have sequential integers 1, 2, 3, …, m in order for a partition to repeat itself in this fashion.

From that, then, it seems the only n that will lead to fixed or stable patterns are those n which satisfy n = (m^2+m)/2 = t(m) [where t(m) is the mth triangular number]. If you check the results which Robert Hood posted to the group in an earlier message, you’ll find that the only groups for which he found a self-repeating partition were for 3, 6, 10, 15, 21, 28 and 36 goats. Or, in other words, where the total number—3 (1+2), 6 (1+2+3), 10 (1+2+3+4), 15 (1+2+3+4+5), 21 (1+2+3+4+5+6), 28 (1+2+3+4+5+6+7) and 36 (1+2+3+4+5+6+7+8)—is triangular.

Using some custom perl code to generate partition sets for arbitrary n, and the graphviz package to visualize them, I’ve generated a number of images that illustrate the pattern.

The 7 partitions of a herd of 5 sheep.
The 7 partitions of a herd of 5 sheep.
The 11 partitions of a herd of 6 sheep.
The 11 partitions of a herd of 6 sheep.
The 15 partitions of a herd of 7 sheep.
The 15 partitions of a herd of 7 sheep.
The 22 partitions of a herd of 8 sheep.
The 22 partitions of a herd of 8 sheep.
The 30 partitions of a herd of 9 sheep.
The 30 partitions of a herd of 9 sheep.
The 42 partitions of a herd of 10 sheep.
The 42 partitions of a herd of 10 sheep.

The question proper asked about herds of 28 and 26 goats, however, which have 3718 and 2436 partitions respectively. I have also generated image files to demonstrate these but they are far too large to embed in the page. If you wish to see them, here is the diagram for a herd of 28 goats (45959x1577px, 1.5MB) and here is the diagram for a herd of 26 goats (29545×1201, 0.95MB).

#!/usr/bin/perl
$n = shift || 6; @p = @h = (); $c = 0;
&part( 2 * $n, $n, 0 );
$fmt = "svg";  # $fmt can also be set to ps, svg; see dot docs
open( OF, ">$n.dot" );
print OF "digraph HERD$n {\n  node [ shape = plaintext ];\n";

# If N > CPU, dot won't be called; set CPU to whatever
# cut-off value is comfortable for your processor's power
$cpu = 30;

for $i ( 0 .. $#h ) {
    @r = @{ $h[$i] };
    $d = '"{' . join( ',', @r ) . '}"';
    for $idx ( 0 .. $#r ) { $r[$idx]-- }
    $v = $#r + 1;
    push @r, $v;
    @a = sort { $b <=> $a } @r;
    @r = @a;
    while ( $r[$#r] == 0 ) { pop @r }
    print OF "  $d -> " . '"{' . join( ',', @r ) . '}"' . "\n";
 }
print OF " }\n"; close(OF);
system("dot -T$fmt $n.dot -o $n.$fmt") if $n < $cpu;
exit;

sub min { my ( $x, $y ) = @_; return ( $x < $y ) ? $x : $y }

sub rec {
    my $t  = shift;
    my @pa = @p; shift @pa; pop @pa while $#pa > $t;
    @{ $h[$c] } = @pa; $c++;
 }

sub part {
    my ( $nn, $k, $t ) = @_; my $j;
    $p[$t] = $k;
    &rec( $t - 1 ) if ( $nn == $k );
    for ( $j = &min( $k, $nn - $k ); $j > 0; $j-- ) {
        &part( $nn - $k, $j, $t + 1 );
     }
 }

The method for generating integer partitions was adapted from NumPartMaxk.c by Dr. Joe Sawada.

Obstinate numbers in the Triangle of the Gods

February 18th, 2003

Combining the earlier work on The Triangle of the Gods problem with the issue of Obstinate numbers has yielded a number of new results, including increasing the range of the largest known obstinate numbers. Specifically, when searching additional lines of the Triangle, I opted to include a test for obstinacy where the number was larger than the currently largest known ON.

For the natural sequence, this yielded:


Row 249:
  1234567891011121314151617181920212223242526272829303132333435 
  3637383940414243444546474849505152535455565758596061626364656 
  6676869707172737475767778798081828384858687888990919293949596 
  9798991001011021031041051061071081091101111121131141151161171 
  1811912012112212312412512612712812913013113213313413513613713 
  8139140141142143144145146147148149150151152153154155156157158 
  1591601611621631641651661671681691701711721731741751761771781 
  7918018118218318418518618718818919019119219319419519619719819 
  9200201202203204205206207208209210211212213214215216217218219 
  2202212222232242252262272282292302312322332342352362372382392 
  40241242243244245246247248249

Row 265:
  1234567891011121314151617181920212223242526272829303132333435 
  3637383940414243444546474849505152535455565758596061626364656 
  6676869707172737475767778798081828384858687888990919293949596 
  9798991001011021031041051061071081091101111121131141151161171 
  1811912012112212312412512612712812913013113213313413513613713 
  8139140141142143144145146147148149150151152153154155156157158 
  1591601611621631641651661671681691701711721731741751761771781 
  7918018118218318418518618718818919019119219319419519619719819 
  9200201202203204205206207208209210211212213214215216217218219 
  2202212222232242252262272282292302312322332342352362372382392 
  4024124224324424524624724824925025125225325425525625725825926 
  0261262263264265

Row 267:
  1234567891011121314151617181920212223242526272829303132333435 
  3637383940414243444546474849505152535455565758596061626364656 
  6676869707172737475767778798081828384858687888990919293949596 
  9798991001011021031041051061071081091101111121131141151161171 
  1811912012112212312412512612712812913013113213313413513613713 
  8139140141142143144145146147148149150151152153154155156157158 
  1591601611621631641651661671681691701711721731741751761771781 
  7918018118218318418518618718818919019119219319419519619719819 
  9200201202203204205206207208209210211212213214215216217218219 
  2202212222232242252262272282292302312322332342352362372382392 
  4024124224324424524624724824925025125225325425525625725825926 
  0261262263264265266267

Row 299:
  1234567891011121314151617181920212223242526272829303132333435 
  3637383940414243444546474849505152535455565758596061626364656 
  6676869707172737475767778798081828384858687888990919293949596 
  9798991001011021031041051061071081091101111121131141151161171 
  1811912012112212312412512612712812913013113213313413513613713 
  8139140141142143144145146147148149150151152153154155156157158 
  1591601611621631641651661671681691701711721731741751761771781 
  7918018118218318418518618718818919019119219319419519619719819 
  9200201202203204205206207208209210211212213214215216217218219 
  2202212222232242252262272282292302312322332342352362372382392 
  4024124224324424524624724824925025125225325425525625725825926 
  0261262263264265266267268269270271272273274275276277278279280 
  281282283284285286287288289290291292293294295296297298299

Row 327:
  1234567891011121314151617181920212223242526272829303132333435 
  3637383940414243444546474849505152535455565758596061626364656 
  6676869707172737475767778798081828384858687888990919293949596 
  9798991001011021031041051061071081091101111121131141151161171 
  1811912012112212312412512612712812913013113213313413513613713 
  8139140141142143144145146147148149150151152153154155156157158 
  1591601611621631641651661671681691701711721731741751761771781 
  7918018118218318418518618718818919019119219319419519619719819 
  9200201202203204205206207208209210211212213214215216217218219 
  2202212222232242252262272282292302312322332342352362372382392 
  4024124224324424524624724824925025125225325425525625725825926 
  0261262263264265266267268269270271272273274275276277278279280 
  2812822832842852862872882892902912922932942952962972982993003 
  0130230330430530630730830931031131231331431531631731831932032 
  1322323324325326327

For the digit sequence:


Row 517:
  1234567890123456789012345678901234567890123456789012345678901 
  2345678901234567890123456789012345678901234567890123456789012 
  3456789012345678901234567890123456789012345678901234567890123 
  4567890123456789012345678901234567890123456789012345678901234 
  5678901234567890123456789012345678901234567890123456789012345 
  6789012345678901234567890123456789012345678901234567890123456 
  7890123456789012345678901234567890123456789012345678901234567 
  8901234567890123456789012345678901234567890123456789012345678 
  90123456789012345678901234567

Row 537:
  1234567890123456789012345678901234567890123456789012345678901 
  2345678901234567890123456789012345678901234567890123456789012 
  3456789012345678901234567890123456789012345678901234567890123 
  4567890123456789012345678901234567890123456789012345678901234 
  5678901234567890123456789012345678901234567890123456789012345 
  6789012345678901234567890123456789012345678901234567890123456 
  7890123456789012345678901234567890123456789012345678901234567 
  8901234567890123456789012345678901234567890123456789012345678 
  9012345678901234567890123456789012345678901234567

Row 561:
  1234567890123456789012345678901234567890123456789012345678901 
  2345678901234567890123456789012345678901234567890123456789012 
  3456789012345678901234567890123456789012345678901234567890123 
  4567890123456789012345678901234567890123456789012345678901234 
  5678901234567890123456789012345678901234567890123456789012345 
  6789012345678901234567890123456789012345678901234567890123456 
  7890123456789012345678901234567890123456789012345678901234567 
  8901234567890123456789012345678901234567890123456789012345678 
  9012345678901234567890123456789012345678901234567890123456789 
  012345678901

Row 575:
  1234567890123456789012345678901234567890123456789012345678901 
  2345678901234567890123456789012345678901234567890123456789012 
  3456789012345678901234567890123456789012345678901234567890123 
  4567890123456789012345678901234567890123456789012345678901234 
  5678901234567890123456789012345678901234567890123456789012345 
  6789012345678901234567890123456789012345678901234567890123456 
  7890123456789012345678901234567890123456789012345678901234567 
  8901234567890123456789012345678901234567890123456789012345678 
  9012345678901234567890123456789012345678901234567890123456789 
  01234567890123456789012345

The pentalpha

February 2nd, 2002

Some notes on the historical role of the figure in response to someone’s assertion that it “symbolizes both mystical and diabolic aspects of the number 5”.

While the pentagram has a distinct and quite old history, the “diabolic” bit is a relatively modern connotation and typically a misunderstanding. None of the ancient sources consider it so. The earliest source I have seen to remark on any “diabolic” connection of any sort to the symbol of the five-pointed star is Éliphas Lévi and his remarks were strictly confined to the symbol with one point downward. Still, he offers no sources for his claim and there are no known precedents for it, so, like most of his writings, one is left to assume it’s purely an invention of his own mind, not a traditional symbol or concept.

When conceived of as five interlocked majuscule Greek Alpha’s, the pentalpha, the symbol actually played a significant role for the Pythagoreans, and is related as well to the golden ratio. The simplest form of the symbol can be traced as far back as Uruk IV (3500 b.c.e. or so) in Mesopotamia where it seemed to merely indicate any heavenly body. During the cuneiform period (2600 b.c.e. & after), the symbol—then called UB—represented a “region” or “direction”, some particular “heavenly quarter”. It is found frequently on potsherds in Uruk, and more frequently still on Jemdet Nasr and proto-Elamite tables (3000–2500 b.c.e.), though examples elsewhere during those periods are rare. There are also examples of the symbol turning up later (early 5th C. b.c.e) in Attic cups and shards. A series of coins bearing the word PENSU (Etruscan for 5) and the symbol have turned up in Greek, Aryan and Etruscan regions. During the Republic in Rome, the symbol was the sign for the building trades.

In the early Christian period it was frequently employed as a Christian symbol said to signify the five wounds of Christ. In the mediæval period it first began to turn up frequently in magical texts, the so-called Solomonic grimoires in particular, where it was usually called a pentacle/pantacle or pentangle, but most sources seem to agree that the specific terms “pentacle” and “pentangle” do not derive from the Greek “penta” but from an old French term meaning “to hang”, usually indicating a “talisman” of one kind or another for wearing about the neck. This seems to be supported by the fact that the Solomonic texts applied the same word equally across the board to both circles inscribed with the five-pointed figure as to those inscribed with a six-pointed hexagram. Both symbols have been called the “Seal of Solomon” through Jewish, Arabic, and theurgic sources.

The first known usage of the symbol in an English source is apparently in Sir Gawain and the Green Knight, stanzas 27–28 (c. 1380 c.e.) were Gawain carries a shield with

shining gules,
With the Pentangle in pure gold depicted thereon.

[…]

It is a symbol which Solomon conceived once
To betoken holy truth, by its intrinsic right,
For it is a figure which has five points,
And each line overlaps and is locked with another;
And it is endless everywhere, and the English call it,
In all the land, I hear, the Endless Knot.

In Tycho Brahe’s Calendarium Naturale Magicum Perpetuum (1582) the symbol is illustrated with a human body drawn within and the letters Y, H, S, V, H drawn at the points (the Hebrew letters יהשוה which were thought by the early Christian kabbalists—Ficino, Mirandola, Reuchlin, etc.—to signify “Jesus” by adding the letter ש in the center of the tetragram יהוה of the Old Testament texts). A contemporary of Brahe, Agrippa, pupil of the Abbot Trithemius, in his de Occulta Philosophia libri tres (available as vol. 48 in the Studies in the History of Christian Thought series from Brill Academic Publishers) presented a very similar illustration but showing the traditional “planetary” glyphs. Robert Fludd produced a similar symbol at the time, as did da Vinci following the symbolism of the Roman architect Vitruvius. Throughout the period the magical texts used it not as a symbol for “evil” but rather as a symbol intended to keep it away. That same idea crops up in Goethe’s Faust (1808) where the symbol prevents Mephistopheles from entering:

Meph.: Let me own up! I cannot go away;
A little hindrance bids me stay.
The witch’s foot upon your still I see.

Faust: The pentagram? That’s in your way?
You son of Hell explain to me,
If that stays you, how came you in today?
And how was such a spirit so betrayed?

Meph.: Observe it closely! It is not well made;
One angle, on the outer side of it,
is just a little open, as you see.

Biedermann, in his Dictionary of Symbolism, indicates that the pentagram was a Christian symbol all the way up through this period. The Roman Emperor, Constantine I, after defeating Maxentius and issuing the Edict of Milan in 312 c.e., credited his conversion to Christianity for his success and from that point on incorporated the pentagram, one point downward, into his personal seal. And while it was a lesser used Christian symbol than, say, the cross or the stylized fish, and may be arguably “rare” in appearance, there are no appearances of the symbol as representing any kind of “devil” or “fiend” at any known point preceding Lévi’s 19th-century writing. The symbol was still being used in a Christian context as late as 1824 (perhaps later) since it is indented on the gateposts of the churchyard of St. Peter’s in Walworth, England which was built in that year.

It did not become a widely recognized symbol for anything “diabolic” until the late ’60s when LaVey founded his “Church of Satan” and used the one-point-down form as the primary symbol of the same.

Obstinate Numbers

January 27th, 2002

Dr. Pickover commented:

In 1848, Camille Armand Jules Marie, better known as the “Prince de Polignac,” conjectured that every odd number is the sum of a power of two and a prime. (For example 13 = 23+5.) He claimed to have proved this to be true for all numbers up to three million, but de Polignac probably would have kicked himself if he had know that he missed 127, which leaves residuals of 125, 123, 119, 111, 95 and 63 (all composites) when the possible powers of two are subtracted from it. There are another 16 of these odd numbers—which my colleague Andy Edwards calls “obstinate numbers”—that are less than 1000. There are an infinity of obstinate numbers greater than 1000. Most obstinate numbers we have discovered are prime themselves. The first composite obstinate number is 905.

Then he asked:

What is the largest obstinate number you can compute?

Can you find an obstinate number terminating in every digit from 0 to 9, or are certain terminal digits impossible to find?

Other unanswered questions:

1) What is the smallest difference between adjacent obstinates?

2) Do any obstinates undulate? (Undulating numbers are of the form: ababababab… For example, 171717 and 28282 are undulating numbers, but they’re not obstinate, as far as I know.)

If you make discoveries, give us some ideas about the kinds of search programs you used.

Curiosity piqued, I looked into the subject. As far as I can tell, the only upper limit as to the size of potentially “obstinate” numbers would be the amount of computer processing power you’d want to put into it. I ran an extensive search and there was no appreciable lessening to the frequency of the numbers and no suggestion that they would “level off” at any point, so I have no reason to not suppose they are an infinite set. The largest I have found so far is:

999999999999999999999999999999999999999999999999999999999999999 
999999999999999999999999999999999999999999999999999999999999999 
9999999999999999999999999999999999999999999999999999999999037

I stopped there not because I’d reached the limits of my processing power, but only because I’d reached the limit of my patience for waiting for further results. It leaves a composite residue for all 621 possible powers of two than can be subtracted from it. Later, this result increased to:

999999999999999999999999999999999999999999999999999999999999999 
999999999999999999999999999999999999999999999999999999999999999 
999999999999999999999999999999999999999999999999999999999999999 
999999999999999999999999999999999999999999999999999999999999999 
999999999999999999999999999999999999999999999999999999999999999 
999999999999999999999999999999999999999999999999999999999999999 
999999999999999999999999999999999999999999999999999999999999999 
999999999999999999999999999999999999999999999999999999999999999 
9999999011

This leaves 1,708 powers of two in the residue. (This number has since been increased; see the article, Obstinate Numbers in the Triangle of the Gods.)

Dr. Pickover also asked if obstinate numbers could terminate in all ten natural digits. I suspect he was trying to trip one up with that since the very definition of obstinate numbers requires them to be odd—as such, they cannot end with an even digit. All odd digits are accounted for, however, in the results I have so far found. As far as the question about the smallest difference between adjacent obstinates, it has to be two. Both 905 and 907 are obstinate, and since obstinate numbers must be odd, two is the smallest possible difference. I would be interested, myself, in the largest difference, and I will probably run a search for it later, but I haven’t yet. When I do, I’ll update these notes. He asked about undulation, and from what I can see, it doesn’t seem to be at all uncommon with these numbers. At a glance, these undulating numbers are all obstinate:

6161
14141
39393
91919
1313131
1818181
7070707
7474747
7676767
7979797
59595959
73737373
343434343
757575757
797979797
929292929
1717171717
3131313131
9191919191
12121212121
14141414141
18181818181
32323232323
54545454545
78787878787
91919191919
7171717171717
25252525252525
29292929292929
37373737373737
43434343434343
67676767676767
97979797979797

I’m certain there are others; I simply didn’t bother continuing the search at higher numbers.

As far as the question as to what search programs were used, for the above I simply wrote two Maple functions—one to check for obstinacy and another to print a list of composite residues. Once those two were in place, the search was easy—just a matter of waiting. For any interested parties, I’ll reproduce the body of the two functions here. They can probably be optimized, but I’m not an especially adept Maple coder (Maple code reminds me far too much of BASIC and that gives me nightmares).

with(numtheory):
isObstinate := proc(o)
  local pp, p, a, z:
  if o mod 2 <> 0 then
    pp := 0: a := floor(log[2](o)):
    for z to a do
      p := o - 2^z:
      if isprime(p) then pp := 1 fi:
    od:
    if pp = 0 then true else false fi:
  else
    printf("Must be oddn"):
  fi:
end:
showResidue := proc(n)
  local a, z, p:
  a := floor(log[2](n)):
  for z to a do
    p := n - 2^z:
    if not isprime(p) then
      printf("2^%d + %dn",z,p)
    fi:
  od:
end:

With those it’s simply a matter of running through the odd numbers and checking isObstinate(number). If a residue list is desired, just throw an obstinate number to showResidue(number). (Note that the latter function assumes you’re feeding it an obstinate number: it doesn’t check to see if you have, or if all of the residue values printed are composite. With the way I used it, there was no point in having the redundant code so I omitted it.)

Slightly more structured test conditions for Mathematica:


ObstinateQ[n_Integer /; n > 0] :=
  If[OddQ[n],
    If[Intersection[
      Table[PrimeQ[n - 2^j], {j, 1, Floor[Log[2, n]]}],
      {True}] == {},
      True, False
     ],
    Print["Must be odd."]
   ]

NPResidue[n_Integer /; ObstinateQ[n]] :=
  If[
    rt = Table[n - 2^j, {j, 1, Floor[Log[2, n]]}];
    Intersection[Map[PrimeQ, rt], {True}] == {},
    rt, 
    Print["Residue includes Prime(s)."]
   ]

A last note, Dr. Pickover remarked that “most” obstinate numbers they’d found were prime numbers themselves. It leads me to believe that they didn’t research much beyond the 1000 point since from the lowest obstinate number, 127, to 100,000, I found a total of 3,392 obstinate numbers, 2,147 of which were composite: the 1,245 prime values were clearly the minority. Considering the decreasing frequency of primes in general as numbers grow increasingly larger, and the seeming continued regularity of obstinate numbers, it seems that the nonprime values will naturally continue to widen the gap between themselves and their prime counterparts.


The Caduceus

October 8th, 2001

The following mailing list remark prompted some thoughts I thought I’d share:

If anyone wonders about the origin of the Caduceus, it actually goes all the way back to the book of Genesis. Moses had to make a copper serpent for the Israelites to look at when bitten by poisonous snakes (during a plague that God had sent) in order to be cured. More info here: http://www.publicsafe.net/wand.htm

Actually, caducei (or caduceum [a.k.a. caduceus] in the singular; κηρύκειον in the Greek) were simply herald staves, signs of peace which, much like the later “white flag”, allowed the messenger to travel unmolested. In their original form they were nothing more than olive sticks with stemmata (wreaths, garlands, etc.) attached to or twined about them; only in later times did the stemmata develop into a serpent motif in artistic representations. The caduceus was not associated with the Greek Hermes or Roman Mercury until some time later, and then only insofar as Hermes/Mercury was considered to be the messenger of the gods. As near as I can tell, caducei did not become symbols of the medical profession until after WWI, due to the exposure it received then by being the symbol the U.S. Army had adopted (around 1902) to indicate its medical department (which use was itself, as I understand it, based on a mistaken assumption). What mistake would that be? In the early 19th century a medical publisher adopted the symbol as part of his imprint—not because he published medical literature, but because he viewed himself as a deliverer of information: a herald, in other words, such as originally carried a caduceus. The usage in the medical profession can be seen as a mixture of two things: 1) the proliferation and perpetuation of a popular error (i.e., that described above) where others adopt the sign because they have seen it used elsewhere in the same context without realizing that the previous usage was itself incorrect; and 2) a confusion of the caduceus with the staff of the healer/medicine god Asclepius of ancient Greece, which was usually represented as a T-shaped staff with a single snake (the snake related mythologically to Asclepius) wrapped around it, which symbol was used to represent “doctors” even in ancient times. The snake and staff combination can be seen in many of the sculptures of Asclepius; e.g., see http://bellarmine.lmu.edu/faculty/fjust/Asclepius.htm. It can also be seen in the symbol of the Royal Society of Medicine. For more information, you might be interested in having a look at the W.J. Friedlander (Prof. Emeritus of Medical Humanities and Neurology at the U. of Nebraska’s College of Medicine) book The Golden Wand of Medicine which can be found on Amazon.

As to Moses, I take it they are referring to the scene depicted in the Book of Numbers, chapter 21, verse 8. It might, or it might not, interest you to know that the word “serpent”, “snake” or any related term appears nowhere in that verse. According to the Hebrew:

ויאמר יהוה אל־משה עשה לך שרף ושים אתו על־נס והיה כל־הנשוך וראה אתו וחי׃

YHWH said to Moses: make unto you a seraph and set it on a pole so that all who are bitten who look at it shall live.

This term seraph and its plural seraphim appear in seven places in the Tanakh:

YHWH sent seraph serpents against the people. They bit, and many Israelites died.

[The Hebrew נחש, meaning “serpent”, does appear in this verse, though not in 21:8.]

Numbers 21:6

YHWH said to Moses: make unto you a seraph and set it on a pole so that all who are bitten who look at it shall live.

Numbers 21:8

who led you through the great and terrible wilderness with its seraph serpent and scorpions, a parched land with no water in it, who brought forth water for you from the flinty rock;

[The Hebrew for serpent appears again here, but is curiously in the singular.]

Deuteronomy 8:15

Seraphs stood in attendance on Him. Each of them had six wings: with two he covered his face, with two he covered his legs, and with two he would fly.

Isaiah 6:2

Then one of the seraphs flew over to me with a live coal, which he had taken from the altar with a pair of tongs.

Isaiah 6:6

Rejoice not, all Philistia,
Because the staff of him that beat you is broken.
From the stock of a snake there sprouts an asp,
A flying seraph branches out from it.

Isaiah 14:29

The ‘Beasts of the Negeb’ pronouncement:
Through a land of distress and hardship,
Of lion and roaring king-beast,
Of viper and flying seraph,
They convey their wealth on the backs of asses,
Their treasures on camels’ humps,
To a people of no avail.

Isaiah 30:6

At least two of the Isaiah passages, those of chapter six, indicate that seraphim are angels; more particularly, that they are angels concerned with praising YHWH and “cleansing” his servants. E.g., Isaiah 6:3-7, in reference to the seraphim earlier described:

And one would call to the other,
“Holy, holy, holy!
YHWH of Hosts!
His presence fills all the earth!”

The doorposts would shake at the sound of the one who called, and the House kept filling with smoke. I cried,

“Woe is me; I am lost!
For I am a man of unclean lips
And I live among a people
Of unclean lips;
Yet my own eyes have beheld
The King YHWH of Hosts.”

Then one of the seraphs flew over to me with a live coal, which he had taken from the altar with a pair of tongs. He touched it to my lips and declared,

“Now that this has touched your lips,
Your guilt shall depart
And your sin be purged away.”

So from the Isaiah 6 descriptions, we can ascertain that a seraph: has arms, a distinct face, six wings (perhaps paired, but not necessarily so), can sing or chant, can speak to a man, can employ fire or some other burning thing such as a coal (or perhaps a venomous serpent? but see below for more on that) to “cleanse” “sin” (it should probably be noted that seraph itself seems to derive from the Hebrew verb meaning “to burn” [e.g., Isaiah 44:19, saraphti, meaning “I have burned”, etc.], so it seems logical that they would have some connection to fire); we might also suspect that they have hands or arms of some kind since how else would they use “tongs” to retrieve a piece of coal from the altar?

If we assume that Revelations 4:8 is a valid commentary on them, then we may also consider that they are apparently covered with eyes.

In Jewish lore and legend, and also in various derivative Christian sources later, seraphim were considered to be the highest “order” of angels, always surrounding the Throne and chanting the trishagion (“Holy, Holy, Holy”), or kedushah for the Hebrew term. According to the third book of Enoch, there are said to be only four of them, corresponding to the four “parts” of the world. In rabbinic sources, they are often equated with the chayyot—the “living creatures” described in the Merkavah literature and the visions of Ezekiel. According to the second book of Enoch, in addition to the six wings mentioned in Isaiah, the seraphim are said to have four faces (like the chayyot). The pseudepigraphal Revelation of Moses includes the reference: “there came one of the six-winged seraphim, and hurried Adam to the Acherusian lake, and washed him in presence of God” which, again, seems to indicate the sin-purging function (although, curiously, this time with water).

How do they relate to the serpents mentioned? Hard to say, really, and there seems no clear answer. Numbers 21:6 implies that the seraph serpents were sent by YHWH, so one could perhaps consider the angelic appellation to merely be signifying the role of the serpents as a “messenger” of YHWH (since the term “angel” only means “messenger” [which, I suppose, could tie us back into the caduceus topic]); the use of the particular “kind” of angel, the seraphim, could reflect possibly two things: 1) considering the root verb “to burn”, it may simply reflect the venomous or burning character of that particular kind of snake-bite, or 2) it could reflect the role of those snakes as similar to the “sin cleansing” role of the seraphim since the snakes were apparently sent as a punishment for their “sin” of speaking up against their predicament. It wouldn’t be the only case of YHWH sending one or more of his “angels” against people. Considering the Isaiah 14:29 and 30:6 references, they also seem to attribute “flying” to the seraph serpents (it is assumed that serpent is meant in the latter two references since the desert setting mentioned in context doesn’t seem to match the location given for the seraph angels—unless YHWH took his Throne to the desert for a bit of a vacation). I’m afraid I’m not at all knowledgeable about this particular subject, so I’ll ask: is there a venomous species of snake native to such an environment, which leaps or is capable of extreme leaps? (i.e., one that could, in time, through retellings, gain—in the literary context—the mythological capability for “flight”?)

On a tangent, it seems that the seraph Moses was instructed to make in Numbers 21.8 gathered a cult and received a formal name. In II Kings, 18:4, we find:

He abolished the shrines and smashed the pillars and cut down the Asherah. He also broke into pieces the bronze serpent that Moses had made, for until that time the Israelites had been offering sacrifices to it; it was called Nehushtan.

[Nehushtan is simply a concatenation of the two Hebrew words נחש, meaning serpent, and תן, meaning dragon or monster (as used in Ezekiel 32:2, “O great beast among the nations, you are doomed! You are like the dragon in the seas,” etc.), so if we assume the “bronze serpent” to have been real at all, it’s probably a safe assumption that it was of quite a large size. Curiously, “bronze serpent” seems to be a bit of a play on words in the Hebrew, since the adjective translated as “bronze” is written identically like the noun for “serpent” except with an additional final letter: הנחש הנחשת, ha-neHesh ha-neHeshat]

Bewer’s Literature of the Old Testament indicates that the Numbers 21:8 “explanation”, however, was merely a mythopoeic creation of the Elohist author—a bit of popular propaganda, in other words—intended to give an origin to the ancient Nehushtan idol that was at the time being worshipped in the temple. According to Bewer, it was thought that by explaining that the idol didn’t represent any sort of god or the like, but instead had simply been a temporary item created by Moses for a specific purpose—namely the healing of snake-bites—for a specific instance, that people would be dissuaded from further worship; Bewer continues, though, stating that the story ultimately had the opposite effect, causing people to increase sacrifices to it, only then considering it to be capable of healing them, in addition to whatever else they’d attributed to it previously.

Still, though, my point was: there’s no historical connection between the supposed seraph of Moses and the caduceus of Rome. Similarity of symbols doesn’t mean equality of symbols, nor does it imply that one was borrowed or derived from the other. It’s not at all uncommon to find wholly unrelated—though visually similar—symbols spread out across all kinds of disparate locales and cultures.

Triangle of the Gods

September 30th, 2001

Dr. Pickover proposes the problem,

An angel descends to Earth and shows the following simple progression of numbers:

1
12
123
1234
12345
123456
1234567
12345678
… etc.

The angel will let you enter the afterlife if you can address two or more of the questions below in a satisfactory manner:

  1. What is the first and second prime number we encounter on this triangle?

  2. 2) What percentage of prime numbers do you expect as we scan more rows in the mysterious triangle?

  3. If you could add one digit to the beginning of each number in order to increase the number of primes, what would it be? Justify your answer.

  4. If you could add one digit to the end of each number in order to increase the number of primes, what would it be? Justify your answer.

  5. What is the largest prime number in this progression known to humanity?

Given the small sample size of the beginning of the triangle, I was uncertain which of two possible forms to pursue. For instance, is it generated by continuing to add the natrual numbers in sequence—would this continue 123456789, 12345678910, 1234567891011, etc.—or do the lines simply cycle through the single digits so that the next few lines would continue 123456789, 1234567890, 12345678901, 123456789012, etc.? Unsure, I replied to both cases; for convenience, I’ll refer to the first as the “natural” sequence, and the second as the “digit” sequence.

In the natural sequence, I found no prime numbers in the first five hundred thousand rows, yet in the digit sequence, I observed these:


Row 171:
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901

Row 277:
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    1234567890123456789012345678901234567

Row 367:
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    1234567

Row 561:
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901

Row 567:
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567

I found no others in the first 1000 rows of the digit sequence.

For the third question, searching the first two hundred rows of the natural sequence, all digits 1 through 9 except for 5, 6 and 9 produced two primes when prefixed to the rows of the sequence. Six and nine produced one each, and I found none for 5 in this range. For reference, here are the rows and numbers which were prime for the given digits:


Prefixing 1:
  Row   1:
    11
  Row   3:
    1123

Prefixing 2:
  Row  21:
    2123456789101112131415161718192021
  Row  69:
    212345678910111213141516171819202122232425262728293031323334 
    353637383940414243444546474849505152535455565758596061626364 
    6566676869

Prefixing 3:
  Row   1:
    31
  Row  19:
    312345678910111213141516171819

Prefixing 4:
  Row   1:
    41
  Row  17:
    41234567891011121314151617

Prefixing 6:
  Row   1:
    61

Prefixing 7:
  Row   1:
    71
  Row   7:
    71234567
  Row 183:
    712345678910111213141516171819202122232425262728293031323334 
    353637383940414243444546474849505152535455565758596061626364 
    656667686970717273747576777879808182838485868788899091929394 
    959697989910010110210310410510610710810911011111211311411511 
    611711811912012112212312412512612712812913013113213313413513 
    613713813914014114214314414514614714814915015115215315415515 
    615715815916016116216316416516616716816917017117217317417517 
    6177178179180181182183

Prefixing 8:
  Row   3:
    8123
  Row  39:
    812345678910111213141516171819202122232425262728293031323334 
    3536373839

Prefixing 9:
  Row  13:
    912345678910111213

For the first two hundred rows of the digital sequence, prefixing 4 produced the most with 4 primes; 1, 6 and 7 each produced three primes; 3 and 5 each produced two; and 8 and 9 produced one each.


Prefixing 1:
  Row   1:
    11
  Row   3:
    1123
  Row 161:
    112345678901234567890123456789012345678901234567890123456789 
    012345678901234567890123456789012345678901234567890123456789 
    012345678901234567890123456789012345678901

Prefixing 3:
  Row   1:
    31
  Row  11:
    312345678901

Prefixing 4:
  Row   1:
    41
  Row  17:
    412345678901234567
  Row  31:
    41234567890123456789012345678901
  Row  99:
    412345678901234567890123456789012345678901234567890123456789 
    0123456789012345678901234567890123456789

Prefixing 5:
  Row  49:
    51234567890123456789012345678901234567890123456789
  Row  59:
    512345678901234567890123456789012345678901234567890123456789

Prefixing 6:
  Row   1:
    61
  Row  41:
    612345678901234567890123456789012345678901
  Row  71:
    612345678901234567890123456789012345678901234567890123456789 
    012345678901

Prefixing 7:
  Row   1:
    71
  Row   7:
    71234567
  Row  19:
    71234567890123456789

Prefixing 8:
  Row   3:
    8123

Prefixing 9:
  Row  21:
    9123456789012345678901

For the fourth question, from the first two hundred rows of the natural sequence, it would seem that attaching a 1 or a 7 would generate the most: adding 1 as a suffix produces 7 primes, adding 7 produces 8; adding 3 or 9 both produced three each.


1:
  Row   1:
    11
  Row   3:
    1231
  Row   9:
    1234567891
  Row  11:
    12345678910111
  Row  16:
    123456789101112131415161
  Row  26:
    12345678910111213141516171819202122232425261
  Row 114:
    123456789101112131415161718192021222324252627282930313233343 
    536373839404142434445464748495051525354555657585960616263646 
    566676869707172737475767778798081828384858687888990919293949 
    5969798991001011021031041051061071081091101111121131141

3:
  Row   1:
    13
  Row   4:
    12343
  Row  97:
    123456789101112131415161718192021222324252627282930313233343 
    536373839404142434445464748495051525354555657585960616263646 
    566676869707172737475767778798081828384858687888990919293949 
    596973

7:
  Row   1:
    17
  Row   2:
    127
  Row   3:
    1237
  Row   4:
    12347
  Row   5:
    123457
  Row  21:
    1234567891011121314151617181920217
  Row  28:
    123456789101112131415161718192021222324252627287
  Row 107:
    123456789101112131415161718192021222324252627282930313233343 
    536373839404142434445464748495051525354555657585960616263646 
    566676869707172737475767778798081828384858687888990919293949 
    5969798991001011021031041051061077

9:
  Row   1:
    19
  Row  16:
    123456789101112131415169
  Row  22:
    123456789101112131415161718192021229

I found it curious that the first five rows all became prime when attaching the digit 7.

For the digit sequence, the number of primes remains rather close to the same: 1 now produces eight primes, one more than in the natural sequence, and 9 now produces only two, one less than in the natural sequence; 3 and 7 still produce the same number of primes as in the natural sequence (3 and 8 respectively).


1:
  Row   1:
    11
  Row   3:
    1231
  Row   9:
    1234567891
  Row  11:
    123456789011
  Row  17:
    123456789012345671
  Row  19:
    12345678901234567891
  Row  33:
    1234567890123456789012345678901231
  Row 170:
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901
3:
  Row   1:
    13
  Row   4:
    12343
  Row 104:
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012343

7:
  Row   1:
    17
  Row   2:
    127
  Row   3:
    1237
  Row   4:
    12347
  Row   5:
    123457
  Row  41:
    123456789012345678901234567890123456789017
  Row  54:
    1234567890123456789012345678901234567890123456789012347
  Row  91:
    123456789012345678901234567890123456789012345678901234567890 
    12345678901234567890123456789017

9:
  Row   1:
    19
  Row  54:
    1234567890123456789012345678901234567890123456789012349

For the final question, of the largest prime number known: with regard to the natural sequence, I’m not certain, but it would have to be greater than


  123456789101112131415161718192021222324252627282930313233343 
  536373839404142434445464748495051525354555657585960616263646 
  566676869707172737475767778798081828384858687888990919293949 
  596979899100101102103104105106107108109110111112113114115116 
  117118119120121122123124125126127128129130131132133134135136 
  137138139140141142143144145146147148149150151152153154155156 
  157158159160161162163164165166167168169170171172173174175176 
  177178179180181182183184185186187188189190191192193194195196 
  197198199200201202203204205206207208209210211212213214215216 
  217218219220221222223224225226227228229230231232233234235236 
  237238239240241242243244245246247248249250251252253254255256 
  257258259260261262263264265266267268269270271272273274275276 
  277278279280281282283284285286287288289290291292293294295296 
  297298299300301302303304305306307308309310311312313314315316 
  317318319320321322323324325326327328329330331332333334335336 
  337338339340341342343344345346347348349350351352353354355356 
  357358359360361362363364365366367368369370371372373374375376 
  377378379380381382383384385386387388389390391392393394395396 
  397398399400401402403404405406407408409410411412413414415416 
  417418419420421422423424425426427428429430431432433434435436 
  437438439440441442443444445446447448449450451452453454455456 
  457458459460461462463464465466467468469470471472473474475476 
  477478479480481482483484485486487488489490491492493494495496 
  497498499500501502503504505506507508509510511512513514515516 
  517518519520521522523524525526527528529530531532533534535536 
  537538539540541542543544545546547548549550551552553554555556 
  557558559560561562563564565566567568569570571572573574575576 
  577578579580581582583584585586587588589590591592593594595596 
  597598599600601602603604605606607608609610611612613614615616 
  617618619620621622623624625626627628629630631632633634635636 
  637638639640641642643644645646647648649650651652653654655656 
  657658659660661662663664665666667668669670671672673674675676 
  677678679680681682683684685686687688689690691692693694695696 
  697698699700701702703704705706707708709710711712713714715716 
  717718719720721722723724725726727728729730731732733734735736 
  737738739740741742743744745746747748749750751752753754755756 
  757758759760761762763764765766767768769770771772773774775776 
  777778779780781782783784785786787788789790791792793794795796 
  797798799800801802803804805806807808809810811812813814815816 
  817818819820821822823824825826827828829830831832833834835836 
  837838839840841842843844845846847848849850851852853854855856 
  857858859860861862863864865866867868869870871872873874875876 
  877878879880881882883884885886887888889890891892893894895896 
  897898899900901902903904905906907908909910911912913914915916 
  917918919920921922923924925926927928929930931932933934935936 
  937938939940941942943944945946947948949950951952953954955956 
  957958959960961962963964965966967968969970971972973974975976 
  977978979980981982983984985986987988989990991992993994995996 
  997998999100010011002100310041005100610071008100910101011101 
  210131014101510161017101810191020102110221023102410251026102 
  710281029103010311032103310341035103610371038103910401041104 
  210431044104510461047104810491050105110521053105410551056105 
  710581059106010611062106310641065106610671068106910701071107 
  210731074107510761077107810791080108110821083108410851086108 
  710881089109010911092109310941095109610971098109911001101110 
  211031104110511061107110811091110111111121113111411151116111 
  711181119112011211122112311241125112611271128112911301131113 
  211331134113511361137113811391140114111421143114411451146114 
  711481149115011511152115311541155115611571158115911601161116 
  211631164116511661167116811691170117111721173117411751176117 
  711781179118011811182118311841185118611871188118911901191119 
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  712081209121012111212121312141215121612171218121912201221122 
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  212531254125512561257125812591260126112621263126412651266126 
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  212831284128512861287128812891290129112921293129412951296129 
  712981299130013011302130313041305130613071308130913101311131 
  213131314131513161317131813191320132113221323132413251326132 
  713281329133013311332133313341335133613371338133913401341134 
  213431344134513461347134813491350135113521353135413551356135 
  713581359136013611362136313641365136613671368136913701371137 
  213731374137513761377137813791380138113821383138413851386138 
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  714181419142014211422142314241425142614271428142914301431143 
  214331434143514361437143814391440144114421443144414451446144 
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  214631464146514661467146814691470147114721473147414751476147 
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  216431644164516461647164816491650165116521653165416551656165 
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  217031704170517061707170817091710171117121713171417151716171 
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  717481749175017511752175317541755175617571758175917601761176 
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  717781779178017811782178317841785178617871788178917901791179 
  217931794179517961797179817991800180118021803180418051806180 
  718081809181018111812181318141815181618171818181918201821182 
  218231824182518261827182818291830183118321833183418351836183 
  718381839184018411842184318441845184618471848184918501851185 
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  218831884188518861887188818891890189118921893189418951896189 
  718981899190019011902190319041905190619071908190919101911191 
  219131914191519161917191819191920192119221923192419251926192 
  719281929193019311932193319341935193619371938193919401941194 
  219431944194519461947194819491950195119521953195419551956195 
  719581959196019611962196319641965196619671968196919701971197 
  219731974197519761977197819791980198119821983198419851986198 
  719881989199019911992199319941995199619971998199920002001200 
  220032004200520062007200820092010201120122013201420152016201 
  720182019202020212022202320242025202620272028202920302031203 
  220332034203520362037203820392040204120422043204420452046204 
  720482049205020512052205320542055205620572058205920602061206 
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  221232124212521262127212821292130213121322133213421352136213 
  721382139214021412142214321442145214621472148214921502151215 
  221532154215521562157215821592160216121622163216421652166216 
  721682169217021712172217321742175217621772178217921802181218 
  221832184218521862187218821892190219121922193219421952196219 
  721982199220022012202220322042205220622072208220922102211221 
  222132214221522162217221822192220222122222223222422252226222 
  722282229223022312232223322342235223622372238223922402241224 
  222432244224522462247224822492250225122522253225422552256225 
  722582259226022612262226322642265226622672268226922702271227 
  222732274227522762277227822792280228122822283228422852286228 
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  223032304230523062307230823092310231123122313231423152316231 
  723182319232023212322232323242325232623272328232923302331233 
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  723482349235023512352235323542355235623572358235923602361236 
  223632364236523662367236823692370237123722373237423752376237 
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  223932394239523962397239823992400240124022403240424052406240 
  724082409241024112412241324142415241624172418241924202421242 
  224232424242524262427242824292430243124322433243424352436243 
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  224832484248524862487248824892490249124922493249424952496249 
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  225132514251525162517251825192520252125222523252425252526252 
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  226032604260526062607260826092610261126122613261426152616261 
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  726482649265026512652265326542655265626572658265926602661266 
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  727382739274027412742274327442745274627472748274927502751275 
  227532754275527562757275827592760276127622763276427652766276 
  727682769277027712772277327742775277627772778277927802781278 
  227832784278527862787278827892790279127922793279427952796279 
  727982799280028012802280328042805280628072808280928102811281 
  228132814281528162817281828192820282128222823282428252826282 
  728282829283028312832283328342835283628372838283928402841284 
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  728582859286028612862286328642865286628672868286928702871287 
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  728882889289028912892289328942895289628972898289929002901290 
  229032904290529062907290829092910291129122913291429152916291 
  729182919292029212922292329242925292629272928292929302931293 
  229332934293529362937293829392940294129422943294429452946294 
  729482949295029512952295329542955295629572958295929602961296 
  229632964296529662967296829692970297129722973297429752976297 
  729782979298029812982298329842985298629872988298929902991299 
  229932994299529962997299829993000300130023003300430053006300 
  730083009301030113012301330143015301630173018301930203021302 
  230233024302530263027302830293030303130323033303430353036303 
  730383039304030413042304330443045304630473048304930503051305 
  230533054305530563057305830593060306130623063306430653066306 
  730683069307030713072307330743075307630773078307930803081308 
  230833084308530863087308830893090309130923093309430953096309 
  730983099310031013102310331043105310631073108310931103111311 
  231133114311531163117311831193120312131223123312431253126312 
  731283129313031313132313331343135313631373138313931403141314 
  231433144314531463147314831493150315131523153315431553156315 
  731583159316031613162316331643165316631673168316931703171317 
  231733174317531763177317831793180318131823183318431853186318 
  731883189319031913192319331943195319631973198319932003201320 
  232033204320532063207320832093210321132123213321432153216321 
  732183219322032213222322332243225322632273228322932303231323 
  232333234323532363237323832393240324132423243324432453246324 
  732483249325032513252325332543255325632573258325932603261326 
  232633264326532663267326832693270327132723273327432753276327 
  732783279328032813282328332843285328632873288328932903291329 
  232933294329532963297329832993300330133023303330433053306330 
  733083309331033113312331333143315331633173318331933203321332 
  233233324332533263327332833293330333133323333333433353336333 
  733383339334033413342334333443345334633473348334933503351335 
  233533354335533563357335833593360336133623363336433653366336 
  733683369337033713372337333743375337633773378337933803381338 
  233833384338533863387338833893390339133923393339433953396339 
  733983399340034013402340334043405340634073408340934103411341 
  234133414341534163417341834193420342134223423342434253426342 
  734283429343034313432343334343435343634373438343934403441344 
  234433444344534463447344834493450345134523453345434553456345 
  734583459346034613462346334643465346634673468346934703471347 
  234733474347534763477347834793480348134823483348434853486348 
  734883489349034913492349334943495349634973498349935003501350 
  235033504350535063507350835093510351135123513351435153516351 
  735183519352035213522352335243525352635273528352935303531353 
  235333534353535363537353835393540354135423543354435453546354 
  735483549355035513552355335543555355635573558355935603561356 
  235633564356535663567356835693570357135723573357435753576357 
  735783579358035813582358335843585358635873588358935903591359 
  235933594359535963597359835993600360136023603360436053606360 
  736083609361036113612361336143615361636173618361936203621362 
  236233624362536263627362836293630363136323633363436353636363 
  736383639364036413642364336443645364636473648364936503651365 
  236533654365536563657365836593660366136623663366436653666366 
  736683669367036713672367336743675367636773678367936803681368 
  236833684368536863687368836893690369136923693369436953696369 
  736983699370037013702370337043705370637073708370937103711371 
  237133714371537163717371837193720372137223723372437253726372 
  737283729373037313732373337343735373637373738373937403741374 
  237433744374537463747374837493750375137523753375437553756375 
  737583759376037613762376337643765376637673768376937703771377 
  237733774377537763777377837793780378137823783378437853786378 
  737883789379037913792379337943795379637973798379938003801380 
  238033804380538063807380838093810381138123813381438153816381 
  738183819382038213822382338243825382638273828382938303831383 
  238333834383538363837383838393840384138423843384438453846384 
  738483849385038513852385338543855385638573858385938603861386 
  238633864386538663867386838693870387138723873387438753876387 
  738783879388038813882388338843885388638873888388938903891389 
  238933894389538963897389838993900390139023903390439053906390 
  739083909391039113912391339143915391639173918391939203921392 
  239233924392539263927392839293930393139323933393439353936393 
  739383939394039413942394339443945394639473948394939503951395 
  239533954395539563957395839593960396139623963396439653966396 
  739683969397039713972397339743975397639773978397939803981398 
  239833984398539863987398839893990399139923993399439953996399 
  739983999400040014002400340044005400640074008400940104011401 
  240134014401540164017401840194020402140224023402440254026402 
  740284029403040314032403340344035403640374038403940404041404 
  240434044404540464047404840494050405140524053405440554056405 
  740584059406040614062406340644065406640674068406940704071407 
  240734074407540764077407840794080408140824083408440854086408 
  740884089409040914092409340944095409640974098409941004101410 
  241034104410541064107410841094110411141124113411441154116411 
  741184119412041214122412341244125412641274128412941304131413 
  241334134413541364137413841394140414141424143414441454146414 
  741484149415041514152415341544155415641574158415941604161416 
  241634164416541664167416841694170417141724173417441754176417 
  741784179418041814182418341844185418641874188418941904191419 
  241934194419541964197419841994200420142024203420442054206420 
  742084209421042114212421342144215421642174218421942204221422 
  242234224422542264227422842294230423142324233423442354236423 
  742384239424042414242424342444245424642474248424942504251425 
  242534254425542564257425842594260426142624263426442654266426 
  742684269427042714272427342744275427642774278427942804281428 
  242834284428542864287428842894290429142924293429442954296429 
  742984299430043014302430343044305430643074308430943104311431 
  243134314431543164317431843194320432143224323432443254326432 
  743284329433043314332433343344335433643374338433943404341434 
  243434344434543464347434843494350435143524353435443554356435 
  743584359436043614362436343644365436643674368436943704371437 
  243734374437543764377437843794380438143824383438443854386438 
  743884389439043914392439343944395439643974398439944004401440 
  244034404440544064407440844094410441144124413441444154416441 
  744184419442044214422442344244425442644274428442944304431443 
  244334434443544364437443844394440444144424443444444454446444 
  744484449445044514452445344544455445644574458445944604461446 
  244634464446544664467446844694470447144724473447444754476447 
  744784479448044814482448344844485448644874488448944904491449 
  244934494449544964497449844994500450145024503450445054506450 
  745084509451045114512451345144515451645174518451945204521452 
  245234524452545264527452845294530453145324533453445354536453 
  745384539454045414542454345444545454645474548454945504551455 
  245534554455545564557455845594560456145624563456445654566456 
  745684569457045714572457345744575457645774578457945804581458 
  245834584458545864587458845894590459145924593459445954596459 
  745984599460046014602460346044605460646074608460946104611461 
  246134614461546164617461846194620462146224623462446254626462 
  746284629463046314632463346344635463646374638463946404641464 
  246434644464546464647464846494650465146524653465446554656465 
  746584659466046614662466346644665466646674668466946704671467 
  246734674467546764677467846794680468146824683468446854686468 
  746884689469046914692469346944695469646974698469947004701470 
  247034704470547064707470847094710471147124713471447154716471 
  747184719472047214722472347244725472647274728472947304731473 
  247334734473547364737473847394740474147424743474447454746474 
  747484749475047514752475347544755475647574758475947604761476 
  247634764476547664767476847694770477147724773477447754776477 
  747784779478047814782478347844785478647874788478947904791479 
  247934794479547964797479847994800480148024803480448054806480 
  748084809481048114812481348144815481648174818481948204821482 
  248234824482548264827482848294830483148324833483448354836483 
  748384839484048414842484348444845484648474848484948504851485 
  248534854485548564857485848594860486148624863486448654866486 
  748684869487048714872487348744875487648774878487948804881488 
  248834884488548864887488848894890489148924893489448954896489 
  748984899490049014902490349044905490649074908490949104911491 
  249134914491549164917491849194920492149224923492449254926492 
  749284929493049314932493349344935493649374938493949404941494 
  249434944494549464947494849494950495149524953495449554956495 
  749584959496049614962496349644965496649674968496949704971497 
  249734974497549764977497849794980498149824983498449854986498 
  74988498949904991499249934994499549964997499849995000

For the digit sequence, unless someone has found one larger, it would be


  Row 567:
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567890123456789012345678901234567890 
    123456789012345678901234567

The next would have to be beyond row 1000, and greater than


  12345678901234567890123456789012345678901234567890123456789012 
  34567890123456789012345678901234567890123456789012345678901234 
  56789012345678901234567890123456789012345678901234567890123456 
  78901234567890123456789012345678901234567890123456789012345678 
  90123456789012345678901234567890123456789012345678901234567890 
  12345678901234567890123456789012345678901234567890123456789012 
  34567890123456789012345678901234567890123456789012345678901234 
  56789012345678901234567890123456789012345678901234567890123456 
  78901234567890123456789012345678901234567890123456789012345678 
  90123456789012345678901234567890123456789012345678901234567890 
  12345678901234567890123456789012345678901234567890123456789012 
  34567890123456789012345678901234567890123456789012345678901234 
  56789012345678901234567890123456789012345678901234567890123456 
  78901234567890123456789012345678901234567890123456789012345678 
  90123456789012345678901234567890123456789012345678901234567890 
  12345678901234567890123456789012345678901234567890123456789012 
  34567890

Arithmology in the Bible

September 26th, 2001

Number, both the concept and specific concrete individual numbers, in the ancient world was a highly philosophized subject, with a variety of treatises written on the symbology of single numbers or whole classes and families of numbers, and much of this material, this arithmology, was taught alongside ordinary mathematics and not as a distinct topic. Elements of these traditions continued down from the Pythagoreans and the neo-Pythagoreans through the kabbalists, Renaissance neo-Platonists, the German paragrammaticists and many others, down through the more diffuse and watered down modern notions of “numerology”.

Often overlooked is the role such number symbolism played in the text of the Bible. (Even the few texts like E.W. Bullinger’s Number in Scripture tend to focus only on the occurrence of literal numbers in the text.) Few seem to notice it at all beyond the ever popular “666” of the Book of Revelation. It’s an example of isopsephia, or Greek gematria, the assigning of numerical values to words or phrases based on the values their constituent letters have in the alphabetic numeral systems of languages such as Hebrew, Arabic, Greek, Coptic and so forth. It was believed that words and expressions whose tallies were equal themselves had some kind of equality or symbolic connection.

῟Ωδε ἡ σοφία ἐστίν. ὁ ἔχων νοῦν ψηφισάτω τὸν ἀριθμὸν τοῦ θηρίου, ἀριθμὸς γὰρ ἀνθρώπου ἐστίν, καὶ ὁ ἀριθμὸς αὐτοῦ ἑξακόσιοι ἑξήκοντα ἕξ.

hic sapientia est qui habet intellectum conputet numerum bestiae numerus enim hominis est et numerus eius est sescenti sexaginta sex

Here is wisdom. Who has understanding, let him calculate the number of the beast, for it is the number of a man, and the number is 666.

Revelation 13:18

Over the ages, an overwhelming amount of time and effort has seemingly been spent by people trying to find a way to make a name—typically of some favored villain of the day—sum to this number, whether the attempts in the earliest days to suggest it was the Emperor Nero to more modern Protestant lines implying it was this Pope or that, all of which seem to miss a slight jest inherent in the problem. What? The Greek phrase τὸ μέγα θηρίον means “the great beast”, and its number is itself 666: τ + ὸ (300 + 70), μ + έ + γ + α (40 + 5 + 3 + 1), θ + η + ρ + ί + ο + ν (9 + 8 + 100 + 10 + 70 + 50), 370 + 49 + 247 = 666. In the same text, there is talk of the woman who rides the beast, arrayed in reds and scarlets; some have observed that ἡ κόκκινη γυνὴ, “the scarlet woman”, enumerates to 667, “one” on “the beast”.

Something slightly strange however is that before the number was in its way demonized via the Revelation reference, 6 and its extension on three scales as 666, was interpreted quite positively. Six is the third of the (arithmologically) important triangular numbers, the last before the tetractys so important in Pythagoreanism. It’s the first perfect number, the first number to be equal to the sum of its aliquot parts; it’s the second circular number; it was held classically to be the number of the “directions” (front/back/left/right or n/s/e/w and up/down) and myriad other things. None of which, of course, mean much to us today as number symbolism holds no great sway in our time, yet to the arithmologists of that era and for a few centuries prior, these thoughts were considered quite important. Pseudo-Iamblichus’ Τα Θεολογούμενα της Αριθμητικής (The Theology of Arithmetic) devotes several pages to the symbolism of the number 6 alone. The next rank of polygonal numbers beyond the triangular is the square, and the square of 6 brings us back around to the number above, in that the square of 6 is 36—and the 36th triangular number is 666, the sum of the numbers from 1 to 36, the sum of 6, 6 by the tetractys and that by the tetractys again. Pseudo-Iamblichus was at pains to observe in that context that 600 is itself the enumeration of κόσμος (world, universe, cosmos). All told, before Revelation 666 should have been positive and important.

Indeed, in Hebrew, the word for gain, profit, advantage, superiority, etc., יתרון, enumerates to 666 (10 + 400 + 200 + 6 + 50), which occurs twelve times in the Old Testament book of Ecclesiastes (1:3; 2:11, 13×2; 3:9; 5:8; 5:15; 7:12; 10:10-11). In I Kings 10:14 and II Chronicles 9:13, it’s stated that Solomon was given 666 talents a year (and the acronym [שמשוך] of that line [שש מאות ששים ושש ככר], itself enumerates to 666), and the number occurs as well as the number of the “sons of Adonikam” in Ezra 2:13 (and their number turns into 667 in the Nehemiah 7:18 version). The number was routinely related to the Sun and solar symbolism, due largely to the ancient numbering of the “planets” associating 6 to the Sun, which made 666 the sum of its magic square.

Adding to the significance is that it is one of the numbers that are “equal in their ranks” (in modern positional notation it simply means that the number has the same digit in the ones, tens, hundreds, etc., places; the multiples of (10^y-1)/9). Virgil, in his Aeneid, ll. 257–277, uses the symbolism of 333; and the Christian numerologists also focused on 777 and 888, where 888 was the enumeration of Ἰησοῦς (Jesus) and by substituting Ϛ (stigma, 6) for the expected στ (s and t, 500) in the word σταυρὸς (cross) it was made to enumerate to 777 (6 + 1 + 400 + 100+ 70 + 200).

The classical theory with gematria (rather than pure arithmology) being, however, that these numbers are less significant as things in themselves than in the context of the words and phrases that bear them as their sum, what do we find if we look for words and phrases enumerating to these numbers? Over the years, I have entertained myself by playing with such number puzzles as these, sometimes doing tallies by hand, sometimes running computerized searches against some text or other. The results are occasionally amusing. Now and then you can run across an older source that suggests one or another of these results was known to earlier generations, and some have been written about elsewhere, but I will present some few oddments here for possible interest.

888

    εἰμι ἡ ζωή (“I am the life”, John 14:6)

    λόγος ἐστί (“the Word is”, which “word” was theologically identified with Jesus)

    παρακληθήσονται (“they shall be comforted”, Matthew 5:4)

    אלהינו עולם ועד

    [“our god forever”, Psalms 48:14/15)]

    אני יהוה לא שניתי

    [“I am YHWH, I do not change”, Malachi 3:6]

    השמים מספרים כבוד-אל

    [“the heavens declare the glory of god”, Psalms 19:2]

    ישועת אלהינו

    [“the salvation of our god”, Psalms 98:3]

777

    τὸν ἄρσενα (“the child”, Revelation 12:13)

    ברקיע השמים

    [“in the expanse of the heavens”, Genesis 1:14]

    שער אור

    [“gate of light”]

    והנחמדים מזהב

    [“more desirable than gold”]

    מרים מגדלית

    [“Mary Magdalene”]

666

    ἀπολλύμεθα (“we perish”, Matthew 8:25, Mark 4:38, Luke 8:24)

    δι’ ἀπιστίαν (“because of disbelief”, Hebrews 3:19)

    εὐπορία (“wealth”)

    παρὰ θεοῦ (“from god”, John 1:6)

    παράδοσις (“tradition”)

    כאלהים

    [“as/like god”]

    אלהיכם

    [“your god”, Isaiah 40:9]

    עשה ארץ

    [“he made the earth”, Jeremiah 10:12, 51:15]

    רקע הארץ

    [“he spread out the earth”, Isaiah 42:5]

    האל יהוה בורא השמים

    [“the god YHWH who made the heavens”, Isaiah 42:5]

    יהי מארת

    [“let there be luminaries”, Genesis 1:14]

    שמש יהוה

    [“YHWH is a sun”, Psalms 84:12/11]

    יום יהוה הוא חשך ולא אור

    [“the day of YHWH will be darkness not light”, Amos 5:18]

    חשיכה ואורה היא כאחד ליהוה

    [“darkness and light are like one to YHWH”, cf. Psalms 139:12]

    ענן וערפל סביבי יהוה

    [“cloud and gloom surround YHWH”, Psalms 97:2]

    סתרו

    [“secret / hiding place”, Psalms 18:12/11]

    מעונך

    [“your dwelling-place”, Psalms 91:9]

    ארון הקדש

    [“holy ark”, II Chronicles 35:3]

    נזר הקדש

    [“holy crown/diadem”, Exodus 29:6, 39:30, Leviticus 8:9]

    משושך

    [“your joy”]

    ורב שלום בניך

    [“and great will be the happiness of your children”, Isaiah 54:13]

    מחרבו הקשה

    [“from his cruel sword”, Isaiah 27:1]

    כעפר הארץ

    [“as the dust of the earth”]

    מעון תנים

    [“den of jackals”, Jeremiah 51:37]

    אין אדם אשר לא יחטא

    [“No man is without sin”, II Chronicles 6:36]

    אני אדם

    [“I am a man”, “I am Adam”]

Integer sequences

September 15th, 2001

On a mailing list, my attention was caught by a comment,

Daniel Dockery, please help me find better sequences.

Unable, or at least unwilling, to resist, I offered the following integer sequence:


                                                                      7
                                                                     47
                                                                    247
                                                                   1247
                                                                   6247
                                                                  31247
                                                                 156247
                                                                 781247
                                                                3906247
                                                               19531247
                                                               97656247
                                                              488281247
                                                             2441406247
                                                            12207031247
                                                            61035156247
                                                           305175781247
                                                          1525878906247
                                                          7629394531247
                                                         38146972656247
                                                        190734863281247
                                                        953674316406247
                                                       4768371582031247
                                                      23841857910156247
                                                     119209289550781247
                                                     596046447753906247
                                                    2980232238769531247
                                                   14901161193847656247
                                                   74505805969238281247
                                                  372529029846191406247
                                                 1862645149230957031247
                                                 9313225746154785156247
                                                46566128730773925781247
                                               232830643653869628906247
                                              1164153218269348144531247
                                              5820766091346740722656247
                                             29103830456733703613281247
                                            145519152283668518066406247
                                            727595761418342590332031247
                                           3637978807091712951660156247
                                          18189894035458564758300781247
                                          90949470177292823791503906247
                                         454747350886464118957519531247
                                        2273736754432320594787597656247
                                       11368683772161602973937988281247
                                       56843418860808014869689941406247
                                      284217094304040074348449707031247
                                     1421085471520200371742248535156247
                                     7105427357601001858711242675781247
                                    35527136788005009293556213378906247
                                   177635683940025046467781066894531247
                                   888178419700125232338905334472656247
                                  4440892098500626161694526672363281247
                                 22204460492503130808472633361816406247
                                111022302462515654042363166809082031247
                                555111512312578270211815834045410156247
                               2775557561562891351059079170227050781247
                              13877787807814456755295395851135253906247
                              69388939039072283776476979255676269531247
                             346944695195361418882384896278381347656247
                            1734723475976807094411924481391906738281247
                            8673617379884035472059622406959533691406247
                           43368086899420177360298112034797668457031247
                          216840434497100886801490560173988342285156247
                         1084202172485504434007452800869941711425781247
                         5421010862427522170037264004349708557128906247
                        27105054312137610850186320021748542785644531247
                       135525271560688054250931600108742713928222656247
                       677626357803440271254658000543713569641113281247
                      3388131789017201356273290002718567848205566406247
                     16940658945086006781366450013592839241027832031247
                     84703294725430033906832250067964196205139160156247
                    423516473627150169534161250339820981025695800781247
                   2117582368135750847670806251699104905128479003906247
                  10587911840678754238354031258495524525642395019531247
                  52939559203393771191770156292477622628211975097656247
                 264697796016968855958850781462388113141059875488281247
                1323488980084844279794253907311940565705299377441406247
                6617444900424221398971269536559702828526496887207031247
               33087224502121106994856347682798514142632484436035156247
              165436122510605534974281738413992570713162422180175781247
              827180612553027674871408692069962853565812110900878906247
             4135903062765138374357043460349814267829060554504394531247
            20679515313825691871785217301749071339145302772521972656247
           103397576569128459358926086508745356695726513862609863281247
           516987882845642296794630432543726783478632569313049316406247
          2584939414228211483973152162718633917393162846565246582031247
         12924697071141057419865760813593169586965814232826232910156247
         64623485355705287099328804067965847934829071164131164550781247
        323117426778526435496644020339829239674145355820655822753906247
       1615587133892632177483220101699146198370726779103279113769531247
       8077935669463160887416100508495730991853633895516395568847656247
      40389678347315804437080502542478654959268169477581977844238281247
     201948391736579022185402512712393274796340847387909889221191406247
    1009741958682895110927012563561966373981704236939549446105957031247
    5048709793414475554635062817809831869908521184697747230529785156247
   25243548967072377773175314089049159349542605923488736152648925781247
  126217744835361888865876570445245796747713029617443680763244628906247
  631088724176809444329382852226228983738565148087218403816223144531247
 3155443620884047221646914261131144918692825740436092019081115722656247
15777218104420236108234571305655724593464128702180460095405578613281247

The last digit remains 7; for numbers of more than one digit, the final two digits remain 47; for numbers with more than two digits, the final three remain 247. For numbers of more than three digits, the fourth-to-the-last digit alternates between 1 and 6 (1 for even x [see below, for how to generate the sequence, to see what I mean by x], 6 for odd). For numbers of more than four digits, the fifth-to-the-last digit repeats the sequence: 3, 5, 8, 0. More than five digits, and the sixth-to-last digit repeats: 1, 7, 9, 5, 6, 2, 4, 0. More than six, the seventh-to-last repeats: 3, 9, 7, 8, 1, 7, 5, 5, 8, 4, 2, 3, 6, 2, 0, 0. And so on.

How is it derived? Let p(n,k) designate the nth polygonal number p of the set of polygonal numbers for a k-sided polygon. E.g., p(n,3) would reveal the triangular numbers, p(n,4) the squares, etc. You can define the function in pari with


p(n,k) = (n/2)*(n*k-k+4-2*n)

or in Maple with


 p := proc(n,k) (n/2)*(n*k-k+4-2*n) end:

The sequence is generated simply by taking p(5,5x)/5 for x=1,2,3…n (n was stopped arbitrarily at 100 in the above example list).

An integer sequence
p(5,5x) ÷ 5 for 1 ≤ x ≤ 120

If you want to skip the polygonal number function altogether, you can render the above sequence with f(x) = 2*5x-3. I included the reference to the polygonal numbers simply to show the source from which I derived the sequence.

In response to the same person having asked about an æsthetically pleasing sequence (to me), I came up with this:


                               1001001
                             101101101101
                          10111111111111101
                        1011112112112112111101
                     101111212212212212212111101
                   10111121222222222222222212111101
                1011112122223223223223223222212111101
              101111212222323323323323323323222212111101
           10111121222232333333333333333333323222212111101
         1011112122223233334334334334334334333323222212111101
      101111212222323333434434434434434434434333323222212111101
    10111121222232333343444444444444444444444434333323222212111101
  1011112122223233334344445445445445445445445444434333323222212111101
101111212222323333434444545545545545545545545545444434333323222212111101
... etc. ...

If you define a generalized function f(x,y) → ((10^x)^y - 1) / (10^x - 1), then you can generate the above by evaluating f(3,y) × f(2,y-1) for successive values of y > 2.

In case the “jumble” of numbers aren’t very clear for some, I rendered a quick image of the sequence colored to show the distribution of prime digits: white areas (inside the general triangle shape) represent non-primes, black areas are the digits 0 and 1; primes are in red.

An integer sequence
f(3,y) × f(2,y-1)

Daniel Dockery

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