Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+dy^2$

Details Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+dy^2$ Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+dy^2$

2011-03-22

arxiv.org/abs/1103.4325v9

Mathematics

Number Theory

45 pages. To appear in the book "Number Theory and the Related Topics"

Scientific paper

Given a squarefree positive integer d, we want to find integers (or rational numbers with denominators not divisible by large primes) $a_0,a_1,a_2,...$ such that for sufficiently large primes p the sum $\sum_{k=0}^{p-1}a_k$ is congruent to $x^2-2p$ mod $p^2$ if $4p=x^2+dy^2$ (and 4|x-2 if d=1), and congruent to 0 mod $p^2$ if (-d/p)=-1. In this paper we give a survey of conjectures and results on this topic and point out the connection between this problem and series for $1/\pi$.

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