# Conjecture: Prime Divisibility & First Differences of Stirling & Eulerian Triangles

## Main Question or Discussion Point

CONJECTURE:
Subtract the Absolute Values of the Stirling Triangle (of the first kind) from those of the Eulerian Triangle. When row number is equal to one less than a prime number, then all entries in that row are divisible by that prime number.

Take for instance, row 6 (see below). The differences between Stirling and Euler Entries are:
0, 42, 217,77,-217,-119

Divide each value by 7 and you get...
0, 6, 31, 11, -31, -17

Note: Row numbers designations are callibrated to n!/(n-1)!, where n! is the row sum....

Stirling Triangle of First Kind (positive and negative signs not shown...)
http://oeis.org/A094638
http://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind

1 --> Row 1
1 01 --> Row 2
1 03 002 --> Row 3
1 06 011 006 --> Row 4
1 10 035 050 024 --> Row 5
1 15 085 225 274 120 --> Row 6

Euler's Triangle (without 0-th row = 1 = 0!)
http://oeis.org/A008292
http://noticingnumbers.net/230EULERStriangle.htm [Broken]

1 --> Row 1
1 01 --> Row 2
1 04 001 --> Row 3
1 11 011 001 --> Row 4
1 26 066 026 001 --> Row 5
1 57 302 302 057 001 --> Row 6

I have only checked this (by hand, not by computer) to Row 11 (more than a year ago). Why? Because I have been trying to look at number progressions (and matrices) as if I were living in the time of Euler, Gauss, etc.. The general hypothesis is that A) one can "discover" meaningful mathematics via observation, a general understanding of how various number progressions relate to one another, and a healthy dose of inductive logic backed by "mathematical facts," even if that "one" be a non-mathematician; and B) that such observations may be based upon very small sample sizes.

A few relevant points:

I. Both triangles are generated via recourse to Binomial Coefficients.

II. All entries in row p Pascal's Triangle of Pascal's Triangle, save the first and and last entries (both 1's), are divisible by p (for p a prime number).

III. The form p-1 figures prominently in both the Euler Totient Function and Wilson's Theorem.

A counter-example or lower bound to this conjecture, or better yet, a proof, would be most welcome. And I am not tied here to being "right." In fact, I would be far more surprised and intrigued should this conjecture prove false.

Best,
Raphie

P.S. The Stirling Triangle of the First Kind is quite well known as it gives the coefficients of n-hedral generating polynomials. Euler's Triangle is less well known, but conceivably important if Frampton & Kephart were on the right track, even if not "right," in their 1999 paper:

Mersenne Primes, Polygonal Anomalies and String Theory Classification
http://arxiv.org/abs/hep-th/9904212

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No responses to this even as the entire forum, or so it seems, bands together (myself included) to protect the integrity of calculus? .999.... equals 1? (See: https://www.physicsforums.com/showthread.php?t=484046)

Someone with proper software and applicable knowledge could either extend the lower bound or refute this conjecture outright with a minimum of effort...

Best,
Raphie

P.S. Where's CRGreathouse when you "need"/want him?

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