Super congruences and elliptic curves over $\Bbb F_p$

Details Super congruences and elliptic curves over $\Bbb F_p$ Super congruences and elliptic curves over $\Bbb F_p$

2010-11-30

arxiv.org/abs/1011.6676v5

Mathematics

Number Theory

20 pages. Add new results and update the references

Scientific paper

In this paper we deduce some new super-congruences motivated by elliptic curves over $\Bbb F_p=\Z/p\Z$. For example, we show that if $p>3$ is a prime and $d\in\{0,1,...,(p-1)/2\}$ then $$\sum_{k=0}^{(p-1)/2}\f{\bi{2k}k\bi{2k}{k+d}}{16^k} =(\f{-1}p)+p^2\f{(-1)^d}4E_{p-3}(d+1/2)\pmod{p^3},$$ where $E_{p-3}(x)$ denotes the Euler polynomial of degree $p-3$. We also raise several new conjectures relating congruences to representations of primes by binary quadratic forms.

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