Abstract
Let \(p>3\) be a prime and let a be a positive integer. We show that if 
or \(a>1\), then
with \((-)\) the Jacobi symbol, which confirms a conjecture of Z.-W. Sun. We also establish the following new congruences:
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Ahlgren, S.: Gaussian hypergeometric series and combinatorial congruences. In: Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (Gainesville, FI, 1999), pp. 1–12, Dev. Math., Vol. 4, Kluwer, Dordrecht (2001)
Lehmer, E.: On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson. Ann. Math. 39, 350–360 (1938)
Mortenson, E.: A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function. J. Number Theory 99, 139–147 (2003)
Mortenson, E.: Supercongruences for truncated \({}_{n+1}F_n\) hypergeometric series with applications to certain weight three newforms. Proc. Am. Math. Soc. 133, 321–330 (2005)
Pan, H., Sun, Z.-W.: A combinatorial identity with application to Catalan numbers. Discret. Math. 306, 1921–1940 (2006)
Pan, H., Sun, Z.-W.: Proof of three conjectures on congruences. Sci. China Math. 57, 2091–2102 (2014)
Ribenboim, P.: The Book of Prime Number Records, 2nd edn. Springer, New York (1989)
Rodriguez-Villegas, F.: Hypergeometric families of Calabi-Yau manifolds, In: Calabi-Yau Varieties and Mirror Symmetry (Toronto, ON, 2001), pp. 223–231, Fields Institute Communications, 38, American Mathematical Society, Providence, RI (2003)
Sun, Z.-H.: Super congruences involving Bernoulli polynomials. Int. J. Number Theory 12, 1259–1271 (2016)
Sun, Z.-W.: A congruence for primes. Proc. Am. Math. Soc. 123, 1341–1346 (1995)
Sun, Z.-W.: Binomial coefficients, Catalan numbers and Lucas quotients. Sci. China Math. 53, 2473–2488 (2010)
Sun, Z.-W.: Super congruences and Euler numbers. Sci. China Math. 54, 2509–2535 (2011)
Sun, Z.-W.: Supercongruences involving products of two binomial coefficients. Finite Fields Appl. 22, 24–44 (2013)
Sun, Z.-W.: Fibonacci numbers modulo cubes of primes. Taiwan. J. Math. 17, 1523–1543 (2013)
Sun, Z.-W.: \(p\)-adic congruences motivated by series. J. Number Theory 134, 181–196 (2014)
Sun, Z.-W., Tauraso, R.: On some new congruences for binomial coefficients. Int. J. Number Theory 7, 645–662 (2011)
Sury, B., Wang, T.-M., Zhao, F.-Z.: Identities involving reciprocals of binomial coefficients. J. Integer Seq. 7(2), Article 04.2.8 (2004)
Wolstenholme, J.: On certain properties of prime numbers. Q. J. Appl. Math. 5, 35–39 (1862)
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The authors would like to thank Prof. Hao Pan and the anonymous referee for helpful comments.
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This research was supported by the Natural Science Foundation of China (Grant 11571162) and the NSFC-RFBR Cooperation and Exchange Program (Grant 11811530072).
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Mao, GS., Sun, ZW. New congruences involving products of two binomial coefficients. Ramanujan J 49, 237–256 (2019). https://doi.org/10.1007/s11139-018-0089-5
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DOI: https://doi.org/10.1007/s11139-018-0089-5