Congruences for Sums of Binomial Coefficients
نویسندگان
چکیده
Let m > 0 and q > 1 be relatively prime integers. We find an explicit period ν m (q) such that for any integers n 0 and r we have n + ν m (q) r m (a) ≡ n r m (a) (mod q), provided that a = −1 and n = 0, or a is an integer with 1 − (−a) m relatively prime to q, where n r m (a) = k≡r (mod m) n k a k. This is a further extension of a congruence of Glaisher.
منابع مشابه
Some Congruences for Central Binomial Sums Involving Fibonacci and Lucas Numbers
We present several polynomial congruences about sums with central binomial coefficients and harmonic numbers. In the final section we collect some new congruences involving Fibonacci and Lucas numbers.
متن کاملOn Sums Related to Central Binomial and Trinomial Coefficients
A generalized central trinomial coefficient Tn(b, c) is the coefficient of x in the expansion of (x+bx+c) with b, c ∈ Z. In this paper we investigate congruences and series for sums of terms related to central binomial coefficients and generalized central trinomial coefficients. The paper contains many conjectures on congruences related to representations of primes by certain binary quadratic f...
متن کاملOn Sums Involving Products of Three Binomial Coefficients
In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of terms involving products of three binomial coefficients. Let p > 3 be a prime. We prove that
متن کاملJacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients
In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence Sn = ( 3n)( 3n 2n) 2( n )(2n+1) , and the binomial coefficients ( 3n n )
متن کاملCongruences for a Class of Alternating Lacunary Sums of Binomial Coefficients
An 1876 theorem of Hermite, later extended by Bachmann, gives congruences modulo primes for lacunary sums over the rows of Pascal’s triangle. This paper gives an analogous result for alternating sums over a certain class of rows. The proof makes use of properties of certain linear recurrences.
متن کاملStatement Julian
My mathematical research interests are in number theory and algebraic geometry. My thesis work concerns the arithmetic of a family of rational numbers known as multiple harmonic sums, which are truncated approximations of multiple zeta values. I explore new structures underlying relations involving multiple harmonic sums, p-adic L-values, Bernoulli numbers, and binomial coefficients. This is us...
متن کامل