A Conjecture concerning the Fibonomial Triangle

TL;DR

This paper investigates the structure of the fibonomial triangle modulo prime numbers, introduces a new proof technique for divisibility patterns, and proposes a conjecture for general prime divisibility based on Fibonacci entry points.

Contribution

It presents a novel proof method for fibonomial divisibility by 5 and formulates a conjecture for prime divisibility conditions based on Fibonacci entry points.

Findings

Fibonomial triangle mod 2 and 3 has a fractal structure.

The triangle exhibits recurring divisibility patterns under mod 5.

Necessary conditions for prime divisibility involve Fibonacci entry points.

Abstract

The fibonomial triangle has been shown by Chen and Sagan to have a fractal nature mod 2 and 3. Both these primes have the property that the Fibonacci entry point of $p$ is $p+1$. We study the fibonomial triangle mod 5, showing with a theorem of Knuth and Wilf that the triangle has a recurring structure under divisibility by five. While this result is not new, our method of proof is new and suggests a conjecture for the divisibility of a fibonomial coefficient by a general prime $p$. We give necessary conditions for such primes, namely that the Fibonacci entry point must be greater than or equal to $p$, and offer numerical evidence for the validity of the conjecture. Lastly, we conclude with a discussion concerning further directions of research.