Proof of a conjecture of Adamchuk
Introduction
In 2006, Adamchuk [1] conjectured that for any prime , After that, many people studied congruences for sums of binomial coefficients (see, for instance, [3], [5], [8], [13], [14], [15], [16], [17], [21], [25], [26], [27]). For example, Pan and Sun [21] used a combinatorial identity to deduce that if p is prime then where is the Jacobi symbol. Then in 2011, Sun and Tauraso [27] gave a generalization of the above result: For any odd prime p and , , Apagodu and Zeilberger [3] proved that for any prime and any positive integer r, Pan and Sun [21, Corollary 1.3] also showed that for any odd prime p, In 2018, Apagodu [2] gave the following conjecture: For any odd prime p, Then the author and Cao [14] confirmed this conjecture and showed another congruence: For any odd prime p, Recall that the Bernoulli numbers and the Bernoulli polynomials are defined as follows: Mattarei and Tauraso [18] proved that for any prime , we have Many researchers also studied numerous complicated congruences involving binomial coefficients and q-binomial coefficients (for instance, [6], [7], [9], [11]).
The main objective of this paper is to obtain the following result which contains the conjecture of Adamchuk. Theorem 1.1 Let p be an odd prime and let . If and , then Theorem 1.2 For any prime , we have
Section snippets
Proof of Theorem 1.2
Define the hypergeometric series where and For a prime p, let denote the ring of all p-adic integers and let For each , define the p-adic order and the p-adic norm . Define the p-adic gamma function by and In particular, we set .
Proof of Theorem 1.1
Proof of Theorem 1.1 Now , so , by (1.1) we have Thus we only need to prove that Let k and l be positive integers with and . In view of the first two congruences in page 7 of [22], we have and So we have Hence we only need to show that It is easy to see
Acknowledgements
The author would like to thank the anonymous referees for helpful comments. This work is funded by the National Natural Science Foundation of China (12001288) and the Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology (2019r062), and it is partially supported by the National Natural Science Foundation of China (12071208).
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