Linear Congruences and a Conjecture of Bibak

We address three questions posed by Bibak {KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences ∑ᵢ₌₁ᵏ aᵢ xᵢ ≡ b 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.