Proof of some conjectural congruences involving products of two binomial coefficients

In this paper, we mainly prove the following conjectures of Z.-W. Sun: Let p equivalent to 3 (mod 4) be a prime. Then p Sigma -1 k=0 p Sigma-1 3 k=0 (2k )2 (2 ) k ) p + 1 (2k- 1)8k equivalent to - ((p + 1)/2 p 2p-1 + 1 (p + 1)/4 (mod p2), (2k )( 2k ) k (2 ) 2p k+1 (2k - 1)8k equivalent to p + ) (mod p2), p ((p+1)/2 (p+1)/4 ) where ( p<middle dot> stands for the Legendre symbol. The necessary proofs are provided by the computer algebra software Sigma to find and verify the underlying hypergeometric sum identities. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.