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

In summary, the conversation discusses a conjecture that subtracting the absolute values of entries in the Stirling Triangle of the first kind from those in the Eulerian Triangle yields a result divisible by the row number, except for the first and last entries. This is based on observations of number progressions and binomial coefficients, and a counter-example or proof is requested. The Stirling Triangle is used for generating polynomials, while Euler's Triangle may be important in string theory classification. The conversation also mentions a separate discussion on the topic of 0.999... being equal to 1, and the potential for someone with proper software and knowledge to extend the lower bound or refute the conjecture.
  • #1
Raphie
151
0
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.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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  • #2


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...Raphie

P.S. Where's CRGreathouse when you "need"/want him?
 
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What is the conjecture about prime divisibility and first differences of Stirling and Eulerian triangles?

The conjecture is that for any prime number p, the first differences of the Stirling and Eulerian triangles modulo p are always divisible by p.

What are Stirling and Eulerian triangles?

Stirling and Eulerian triangles are two types of triangular arrays that are used in combinatorics and number theory. They are named after mathematicians James Stirling and Leonhard Euler, respectively.

What are first differences in a triangular array?

In a triangular array, the first differences refer to the numbers obtained by subtracting each element from the one directly above it. For example, in the Stirling triangle, the first differences of the second row would be 1, 2, 6, 24, 120, ...

Why is this conjecture important?

If proven to be true, this conjecture would have significant implications in the study of prime numbers and the relationships between different types of triangular arrays. It could also potentially lead to new discoveries and insights in mathematics.

What progress has been made towards proving this conjecture?

So far, no one has been able to prove or disprove this conjecture. However, there have been some partial results and attempts at proofs by various mathematicians. This conjecture remains an open problem in mathematics.

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