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Linear congruences and a conjecture of Bibak

Chinnakonda Gnanamoorthy Karthick Babu, Ranjan Bera, Balasubramanian Sury (2024)

Czechoslovak Mathematical Journal

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...

Linear recurrence sequences without zeros

Artūras Dubickas, Aivaras Novikas (2014)

Czechoslovak Mathematical Journal

Let a d - 1 , ⋯ , a 0 ∈ ℤ , where d ∈ ℕ and a 0 ≠ 0 , and let X = ( x n ) n = 1 ∞ be a sequence of integers given by the linear recurrence x n + d = a d - 1 x n + d - 1 + ⋯ + a 0 x n for n = 1 , 2 , 3 , ⋯ . We show that there are a prime number p and d integers x 1 , ⋯ , x d such that no element of the sequence X = ( x n ) n = 1 ∞ defined by the above linear recurrence is divisible by p . Furthermore, for any nonnegative integer s there is a prime number p ≥ 3 and d integers x 1 , ⋯ , x d such that every element of the sequence X = ( x n ) n = 1 ∞ defined as above modulo p belongs to the set { s + 1 , s + 2 , ⋯ , p - s - 1 } .

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