Divisibility Conditions for Fibonomial Coefficients

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 theorem relating the divisibility by a general prime $p$ of a fibonomial coefficient to the divisibility by $p$ of a product of fibonomial coefficients in the first $p$ rows of the fibonomial triangle. This product is constructed using a particular base. We give necessary and sufficient conditions for which primes $p$ satisfy the theorem, namely that the Fibonacci entry point of $p$ must be greater than or equal to $p$ for the theorem to hold. Lastly, we conclude with a discussion concerning further directions of research.

Grantor dc:publisher

Wake Forest University

Year dc:date.issued

2016

Author dc:creator

• Southwick, Jeremiah T.

• Fibonomial

Language dc:language.iso

en

Handle dc:identifier.uri

http://hdl.handle.net/10339/59323

OAI identifier oai:identifier

oai:wakespace.lib.wfu.edu:10339/59323