A REFINEMENT OF A CONGRUENCE RESULT BY VAN HAMME AND MORTENSON

摘要

Let p be an odd prime. In 2008, E. Mortenson proved van Hamme's following conjecture:rn(p-1)/2 Σ k=0 (4k+1)(-1/2 k)~3(=)(-1)~((p-1)/2)p(mod p~3).rnIn this paper, we show further thatrnp-1Σ k=0(4k+1)(-1/2 k)~3(=)(p-1)/2Σk=0(4k+1)(-1+2 k)~3 (=)(-1)~((p-1)/2)p+p~3 E_(p-3)(mod p~4)rnwhere E_0,E_1,E_2,... are Euler numbers. We also prove that if p ＞ 3 thenrn(p-1)/2Σk=0 20k+3/(-2~(10))~k(4k k,k,k,k) (=)(-1)~((p-1)/2)p(2~(p-1)+2-(2~(p-1)-1)~2)(mod p~4).