Abstract
The aim of this article is to use the fundamental modus and the properties of the Euler polynomials and Bernoulli polynomials to prove some new congruences related to Bernoulli polynomials. One of them is that for any integer h or any non-negative integer n, we obtain the congruence B 2 n + 1 ( 2 h ) ≡ 0 mod ( 2 n + 1 ) , where B n ( x ) are Bernoulli polynomials.
Highlights
IntroductionFor the real number x, if m ≥ 0 denotes any integer, the famous Bernoulli polynomials
As usual, for the real number x, if m ≥ 0 denotes any integer, the famous Bernoulli polynomialsBm ( x ) and Euler polynomials Em ( x ) are decided by the coefficients of the series of powers: ∞ z · ezx Bm ( x ) m = ·z ∑ z e −1 m! m =0 (1)
From Definitions 1 and 2 of the Euler polynomials and Bernoulli polynomials, we discover the identity as below:
Summary
For the real number x, if m ≥ 0 denotes any integer, the famous Bernoulli polynomials. Many scholars have studied the properties of these polynomials and numbers, and they have obtained some valuable research conclusions. For any positive integers m and h, the following identity should be obtained, that is:. For any integer h, we obtain the congruence: B2m+1 (2h) ≡ 0 mod (2m + 1), where a b. Let p be an odd prime; as a result, we have:. In this way, there exits an integer N with N ≡ 1 mod p such that the polynomial congruence:.
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