On congruences related to central binomial coefficients
Copyright policy )On congruences related to central binomial coefficients
It is known that $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)4^k)=��/2$ and $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)16^k)=��/3$. In this paper we obtain their p-adic analogues such as $$\sum_{p/23 is a prime and E_0,E_1,E_2,... are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we have $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=4x^2-2p (mod p^2)$$ if (p/7)=1 and p=x^2+7y^2 with x,y integers, and $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=0 (mod p^2)$$ if (p/7)=-1, i.e., p=3,5,6 (mod 7).
- Nanjing University China (People's Republic of)
- Hebei University China (People's Republic of)
- Nanjing University China (People's Republic of)
- Nanjing University China (People's Republic of)
- Nanjing University China (People's Republic of)
Central binomial coefficients, Algebra and Number Theory, Mathematics - Number Theory, binary quadratic forms, central binomial coefficients, congruences modulo prime powers, Binary quadratic forms, Congruences modulo prime powers, Binomial coefficients; factorials; \(q\)-identities, 11B65, 11A07, 05A10, 11B68, 11E25, FOS: Mathematics, Mathematics - Combinatorics, Congruences; primitive roots; residue systems, Number Theory (math.NT), Combinatorics (math.CO), Bernoulli and Euler numbers and polynomials, Sums of squares and representations by other particular quadratic forms, Euler numbers, Factorials, binomial coefficients, combinatorial functions
Central binomial coefficients, Algebra and Number Theory, Mathematics - Number Theory, binary quadratic forms, central binomial coefficients, congruences modulo prime powers, Binary quadratic forms, Congruences modulo prime powers, Binomial coefficients; factorials; \(q\)-identities, 11B65, 11A07, 05A10, 11B68, 11E25, FOS: Mathematics, Mathematics - Combinatorics, Congruences; primitive roots; residue systems, Number Theory (math.NT), Combinatorics (math.CO), Bernoulli and Euler numbers and polynomials, Sums of squares and representations by other particular quadratic forms, Euler numbers, Factorials, binomial coefficients, combinatorial functions
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