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...hat and ( )( ) 2k 3k = k −1 k −1 ( )( ) 2k 3k = k +1 k +1 Mathematica yields that ∞∑ k=0 (( ) 2k −Ck k (( ) )( 2k −Ck 2 k Ck ¯ C (2) k 27 k = 81√ 3 4π −9. ) ( ) 3k (2) k −C k 2 ( ) 3k Conjecture A54 (=-=[S-1]-=-). Let p be an odd prime. Then ∑p−1 k=1 2 k k ( ) 3k ≡ −3pq k 2 p (2) (mod p2 ), k − ¯ C (2) ) k . and p−1 ∑ p k=1 1 k2 k( 3k k { 2 0 (mod p ) if p ≡ 1 (mod 4), ) ≡ −3/5 (mod p2 ) if p ≡ 3 (mod 4).Wh...

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...≡ 0 (mod p). Such congruences are concerned with binary quadratic forms. Let p > 5 be a prime. In [18] Zhao, Pan and Sun obtained the congruence∑p−1 k=1 2 k ( 3k k ) ≡ 6 5 ((−1)(p−1)/2−1) (mod p). In =-=[15]-=- Z.W. Sun investigated∑p−1k=0 (3kk )/mk (mod p) for m 6≡ 0 (mod p). He gave explicit congruences in the cases m = 6, 7, 8, 9, 13, −1 4 , 27 4 , 8 3 . Suppose that p > 3 is a prime and k ∈ {0, 1, . . ....

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...be a prime. In [12] Zhao, Pan and Sun proved that p−1∑ k=1 ( 3k k ) 2k ≡ 6 5 ((−1)(p−1)/2 − 1) (mod p). 1The author is supported by the Natural Sciences Foundation of China (grant no. 11371163). 1 In =-=[10]-=- Z.W. Sun investigated ∑p−1 k=0 (3k k ) ak (mod p) for a ∈ Zp. He gave explicit congruences for a = −4, 16 , 17 , 18 , 19 , 113 , 38 , 427 . Suppose that p > 3 is a prime and k ∈ {0, 1, . . . , p − 1}...

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... follows: C (h) k = 1 hk + 1 ( (h+ 1)k k ) = ( (h+ 1)k k ) − h ( (h+ 1)k k − 1 ) (k ∈ N) and C̄ (h) k = h k + 1 ( (h+ 1)k k ) = h ( (h+ 1)k k ) − ( (h+ 1)k k + 1 ) (k ∈ N). 8 ZHI-WEI SUN In [ZPS] and =-=[S09]-=-, the authors gave various congruences involving higherorder Catalan numbers. In particular, Sun [S09] proved that for any prime p > 3 and a ∈ Z+ with 6 | a we have the congruence ∑ 0<k<pa k≡r (mod p−...

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...authors studied congruences involving the binomial coefficient ( 3k k ) . For example, Zhao, Pan and Sun [15] showed that for any prime p > 5, p−1∑ k=1 2k ( 3k k ) ≡ 6 5 ((−1)(p−1)/2 − 1) (mod p). In =-=[13]-=- Z.W. Sun investigated ∑p−1 k=0 ( 3k k ) m−k (mod p) for a prime p > 3 and m 6≡ 0 (mod p). He gave explicit congruences in the cases m = 6, 7, 8, 9, 13,−14 , 427 , 38 . Let Z and N be the sets of inte...