LAUSR

Abstract In this paper we mainly prove the following two conjectures of Z.-W. Sun [10] : For any odd prime p , we have ∑ k = 0 p − 1 P k 8 k ≡ 1 + 2 ( − 1 ) ( p − 1 ) / 2 p 2 E p − 3 ( mod p 3 ) , ∑ k = 0 p − 1 P k 16 k ≡ ( − 1 ) ( p − 1 ) / 2 − p 2 E p − 3 ( mod p 3 ) , where P n = ∑ k = 0 n ( 2 k k ) 2 ( 2 ( n − k ) n − k ) 2 ( n k ) is the n -th Catalan–Larcombe–French number, E n are the Euler numbers which are defined by E 0 = 1 , E n = − ∑ k = 1 ⌊ n / 2 ⌋ ( n 2 k ) E n − 2 k ( n ≥ 1 ) .