Linear Congruences and a Conjecture of Bibak

2024/03/04 by C. G. Karthick Babu, Babu, C. G. Karthick, Ranjan Bera +3
Mathematics · #11A25 #11D79 #11P83 #11T24 #11T55 #Advanced Mathematical Identities #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2403.01923

openalex publication_date 2024/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We address three questions posed by Bibak \citeKB20, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences ∑_i=1^k a_i x_i ≡ b \Modn. 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 x₁ ≥ ⋯ ≥ x_k or, with strict order restrictions x₁> ⋯ > 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.

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