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Linear Congruences and a Conjecture of Bibak

C. G. Karthick Babu, Ranjan Bera, B. Sury

Abstract

We address three questions posed by Bibak \cite{KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences $\sum_{i=1}^k a_i x_i \equiv b \Mod{n}$. In particular, we obtain explicit expressions for the number of solutions where $x_i$'s are squares modulo $n$. In addition, we obtain expressions for the number of solutions with order restrictions $x_1 \geq \cdots \geq x_k$ or, with strict order restrictions $x_1> \cdots > x_k$ in some special cases. In these results, the expressions for the number of solutions involve Ramanujan sums and are obtained using their properties.

Linear Congruences and a Conjecture of Bibak

Abstract

We address three questions posed by Bibak \cite{KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences . In particular, we obtain explicit expressions for the number of solutions where 's are squares modulo . In addition, we obtain expressions for the number of solutions with order restrictions or, with strict order restrictions in some special cases. In these results, the expressions for the number of solutions involve Ramanujan sums and are obtained using their properties.
Paper Structure (10 sections, 14 theorems, 76 equations)

This paper contains 10 sections, 14 theorems, 76 equations.

Table of Contents

  1. Introduction and Statements of Main Results
  2. Distinct Solutions and Schönemann's Theorem
  3. Square solutions - some subtleties
  4. Preliminaries
  5. Discrete Fourier transform
  6. Gauss sums and Ramanujan sums
  7. Generating functions of partition with certain conditions
  8. Square Solutions - proof of Theorem 1
  9. Order restricted congruence - proof of Theorem 3
  10. Strict order restricted congruence - Proof of Theorem 2

Key Result

Theorem 1

Let $S_{n}(b; a_{1}, \dots, a_{k})$ denote the number of square solutions of linear cong. Assume $n$ is an odd positive integer, having a prime factorization $n=p_{1}^{\ell_{1}}\cdots p_{r}^{\ell_{r}}$. Then, we have where

Theorems & Definitions (17)

  • Theorem 1
  • Corollary 1.1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1.2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 7 more