Linear congruences and a conjecture of Bibak

Chinnakonda Gnanamoorthy Karthick Babu; Ranjan Bera; Balasubramanian Sury

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 4, page 1185-1206
  • ISSN: 0011-4642

Abstract

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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 = 1 k a i x i ≡ 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 ≥ ⋯ ≥ x k or with strict order restrictions x 1 > ⋯ > 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.

How to cite

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Babu, Chinnakonda Gnanamoorthy Karthick, Bera, Ranjan, and Sury, Balasubramanian. "Linear congruences and a conjecture of Bibak." Czechoslovak Mathematical Journal 74.4 (2024): 1185-1206. <http://eudml.org/doc/299651>.

@article{Babu2024,
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 $\sum _\{i=1\}^k a_i x_i \equiv b \hspace\{4.44443pt\}(\@mod \; n)$. 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 $x_1 \ge \cdots \ge x_k$ 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.},
author = {Babu, Chinnakonda Gnanamoorthy Karthick, Bera, Ranjan, Sury, Balasubramanian},
journal = {Czechoslovak Mathematical Journal},
keywords = {system of congruence; restricted linear congruence; Ramanujan sum; discrete Fourier transform},
language = {eng},
number = {4},
pages = {1185-1206},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linear congruences and a conjecture of Bibak},
url = {http://eudml.org/doc/299651},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Babu, Chinnakonda Gnanamoorthy Karthick
AU - Bera, Ranjan
AU - Sury, Balasubramanian
TI - Linear congruences and a conjecture of Bibak
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 4
SP - 1185
EP - 1206
AB - 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 $\sum _{i=1}^k a_i x_i \equiv b \hspace{4.44443pt}(\@mod \; n)$. 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 $x_1 \ge \cdots \ge x_k$ 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.
LA - eng
KW - system of congruence; restricted linear congruence; Ramanujan sum; discrete Fourier transform
UR - http://eudml.org/doc/299651
ER -

References

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