Divisibility of Fibonomial coefficients in terms of their digital representations and applications

Abstract

We give a characterization for the integers $ n \geq 1 $ such that the Fibonomial coefficient $ {pn \choose n}_F $ is divisible by $ p $ for any prime $ p \neq 2, 5 $. Then we use it to calculate asymptotic formulas for the number of positive integers $ n \leq x $ such that $ p \mid {pn \choose n}_F $. This completes the study on this problem for all primes $ p $.

Keywords

• fibonomial coefficient
• fibonacci number
• digit
• sum of digits function
• divisibility
• asymptotic
• binomial coefficient