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Supercongruences for central trinomial coefficients

Hao Pan, Zhi-Wei Sun

Abstract

For each $n=0,1,2,\ldots$, the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. Let $p>3$ be a prime, and let $n$ be any positive integer. In 2016, the second author conjectured that the quotient $(T_{pn}-T_n)/(pn)^2$ is always a $p$-adic integer. In this paper, we confirm this conjecture, and further prove that $$\frac{T_{pn}-T_n}{(pn)^2}\equiv\frac{T_{n-1}}6\left(\frac p3\right)B_{p-2}\left(\frac13\right)\pmod p,$$ where $(\frac p3)$ is the Legendre symbol and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.

Supercongruences for central trinomial coefficients

Abstract

For each , the central trinomial coefficient is the coefficient of in the expansion of . Let be a prime, and let be any positive integer. In 2016, the second author conjectured that the quotient is always a -adic integer. In this paper, we confirm this conjecture, and further prove that where is the Legendre symbol and is the Bernoulli polynomial of degree .

Paper Structure

This paper contains 3 sections, 6 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Proofs of Theorem \ref{['Th1.1']} and Corollary \ref{['Cor1.1']}
  3. Proof of Theorem \ref{['Th1.2']}

Key Result

Theorem 1.1

Let $p>3$ be a prime and let $n$ be a positive integer. Then where $(\frac{p}{3})$ is the Legendre symbol and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Corollary 1.1
  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1