2020/10/26 by Wang, Chen, Hu, Dian-Wang · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2010.13638
In this paper, we prove some supercongruences concerning truncated hypergeometric series. For example, we show that for any prime p>3 and positive integer r, ∑k=0pr-1(3k+1)((\frac12)k3)/((1)k3)4k≡ pr+\frac76pr+3Bp-3\pmodpr+4 and ∑k=0(pr-1)/2(4k+1)((\frac12)k4)/((1)k4)≡ pr+\frac76pr+3Bp-3\pmodpr+4, where (x)k=x(x+1)⋯(x+k-1) is the Pochhammer symbol and B0,B1,B2,… are Bernoulli numbers. These two congruences confirm conjectures of Sun [Sci. China Math. 54 (2011), 2509--2535] and Guo [Adv. Appl. Math. 120 (2020), Art. 102078], respectively.