Jacobsthal sums, Legendre polynomials and binary quadratic forms

2012/02/06 by Zhi-Hong Sun, Sun, Zhi-Hong
Mathematics · #05A10 #11A07 #11A15 #11E25 #33C45 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1202.1237

openalex publication_date 2012/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p>3 be a prime and m,n∈\Bbb Z with p\nmid mn. Built on the work of Morton, in the paper we prove the uniform congruence: amp;∑_x=0^p-1(\fracx³+mx+np) ≡ -(-3m)^\fracp-14 ∑_k=0^p-1\binom-\frac 112k\binom-\frac 512k ((4m³+27n²)/(4m³))^k\pmod pamp;\tif 4| p-1, (2m)/(9n)(\frac-3mp)(-3m)^\fracp+14 ∑_k=0^p-1\binom-\frac 112k\binom-\frac 512k ((4m³+27n²)/(4m³))^k\pmod pamp;if 4| p-3, where (\frac ap) is the Legendre symbol. We also establish many congruences for x\pmod p, where x is given by p=x²+dy² or 4p=x²+dy², and pose some conjectures on supercongruences modulo p² concerning binary quadratic forms.

Related

• Congruences concerning Legendre polynomials II
• Congruences concerning Legendre polynomials III
• A new kind of numbers and related congruences
• Super congruences concerning binomial coefficients and Apéry-like numbers
• Congruences concerning binomial coefficients and binary quadratic forms