Geodesic
Linear congruences and a conjecture of Bibak
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 4, pp. 1185-1206 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\sum _{i=1}^k a_i x_i \equiv b \pmod n$. 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.
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 $\sum _{i=1}^k a_i x_i \equiv b \pmod n$. 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.
DOI : 10.21136/CMJ.2024.0151-24
Classification : 11A25, 11D79, 11P83, 11T24, 11T55
Keywords: system of congruence; restricted linear congruence; Ramanujan sum; discrete Fourier transform 
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     title = {Linear congruences and a conjecture of {Bibak}},
     journal = {Czechoslovak Mathematical Journal},
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Babu, Chinnakonda Gnanamoorthy Karthick; Bera, Ranjan; Sury, Balasubramanian. Linear congruences and a conjecture of Bibak. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 4, pp. 1185-1206. doi: 10.21136/CMJ.2024.0151-24

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