Congruences Involving Bernoulli Numbers and Fermat–Euler Quotients

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p > 5 be a prime. In the paper we mainly determine P[p/4] x=1 1 xk (mod p2), p−1 [p/4] (mod p3), Pp−1 k=1 2 k (mod p3) and Pp−1 k=1 2 k2 (mod p2) in terms of Euler and Bernoulli numbers. For example, we have

General Congruences Involving the Bernoulli Numbers.

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New Types of Congruences Involving Bernoulli Numbers and Fermat's Quotient.

make Mp prime are 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257. Comparison of this list with the correct data recorded in the top line of the table presented below shows that Mersenne made five mistakes. p = 67 and 257 do not yield prime values for Mp, and p = 61, 89 and 107 were not included in his list of special primes. With reference to explicit factoring, attention should be called to a val...

Congruences Involving Combinations of the Bernoulli and Fibonacci Numbers.

Congruences involving Bernoulli polynomials

Let {Bn(x)} be the Bernoulli polynomials. In the paper we establish some congruences for Bj(x) (mod p n), where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p-regular functions, the congruences for h(−sp) (mod p) (s = 3, 5, 8, 12) and the sum P k≡r (mod m) p k , where h(d) is the class number of the quadratic field Q(d) of discriminant d...