A Conjecture concerning the Fibonomial Triangle

الأصل · EN

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.