Congruences Involving Combinations of the Bernoulli and Fibonacci Numbers

Abstract

* Work on this paper was done under National Science Foundation Grant 3697. 1 Different sequences of numbers have been called Lucas sequences. We are using the termi

On the Euler and Bernoulli polynomials.

“…It should be pointed out that identities (a) and (e) are contained in a general formula of R. Kelisky [9]. This result follows from (17), Theorem 21, and the Corollary preceding by observing that x 2 -x -l does not divide Reimirk.…”

“…We note at the outset that θ ~ 1. Also, if we put x = -î nto (9) Ei k (i+x) (The various properties of E n (x) used in the following proof will be found in N rlund [11].)…”

“…If in(9) we put x = -^ , we have E* -= [4 · 5 4 * -(3 4 * It remains therefore to show E^ > 4 · 5 4 */(3 4 * + 1) for A > 1. If in(9) we put x = -^ , we have E* -= [4 · 5 4 * -(3 4 * It remains therefore to show E^ > 4 · 5 4 */(3 4 * + 1) for A > 1.…”