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arXiv · 1604.04775

A Conjecture concerning the Fibonomial Triangle

Jeremiah Southwick
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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.

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Jeremiah Southwick. 2016-04-16. A Conjecture concerning the Fibonomial Triangle. https://arxiv.org/abs/1604.04775

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