Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+d{y}^2$

Abstract

Given a squarefree positive integer $d$, we want to find integers (or rational numbers with denominators not divisible by large primes) $a_0,a_1,a_2,\dots$ such that for sufficiently large primes $p$ we have ${\sum }_{k=0}^{p-1}{a}_k\equiv x^2-2p$ (mod $p^2$) if $4p=x^2+d{y}^2$ (and $4\nmid x$ if $d=1$), and ${\sum }_{k=0}^{p-1}{a}_k\equiv 0$ (mod $p^2$) if $(\frac{-d}{p})=-1$. 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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References in corpus (5)

• Open Conjectures on Congruences
• On congruences related to central binomial coefficients
• On sums involving products of three binomial coefficients
• On sums of Apéry polynomials and related congruences
• Congruences involving generalized central trinomial coefficients

Cited by in corpus (2)

• Proof of some conjectures of Z.-W. Sun on congruences for Apery polynomials
• Supercongruences motivated by e