Linear congruences and a conjecture of Bibak

Authors

Chinnakonda Gnanamoorthy Karthick Babu, Indian Statistical Institute Bangalore
Ranjan Bera, Indian Statistical Institute Bangalore
Balasubramanian Sury, Indian Statistical Institute Bangalore

Article Type

Research Article

Publication Title

Czechoslovak Mathematical Journal

Abstract

We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences ∑i=1k=aixi≡b (mod n). In particular, we obtain explicit expressions for the number of solutions, where xi’s are squares modulo n. In addition, we obtain expressions for the number of solutions with order restrictions x1 ⩾ … ⩾ xk or with strict order restrictions x1 > … > xk in some special cases. In these results, the expressions for the number of solutions involve Ramanujan sums and are obtained using their properties.

First Page

1185

Last Page

1206

DOI

10.21136/CMJ.2024.0151-24

Publication Date

12-1-2024

Comments

Open Access; Green Open Access

Recommended Citation

Babu, Chinnakonda Gnanamoorthy Karthick; Bera, Ranjan; and Sury, Balasubramanian, "Linear congruences and a conjecture of Bibak" (2024). Journal Articles. 4884.
https://digitalcommons.isical.ac.in/journal-articles/4884