On solving a restricted linear congruence using generalized Ramanujan\n sums

2017/08/15 by K Vishnu Namboothiri, Namboothiri, K Vishnu · 1 citation
Mathematics · #11A25 #11D79 #11L03 #11P83 #42A16 #Advanced Mathematical Identities #FOS: Mathematics #Graph theory and applications #Mathematical Inequalities and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1708.04505

openalex publication_date 2017/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the linear congruence equation x₁+\…+x_k \≡ b ,(\mod\n n) for b,n\∈\ℤ. By (a,b)_s, we mean the largest\nl^s\∈\ℕ which divides a and b simultaneously. For each d_j|n,\ndefine \C_j,s = 1\≤ x\≤ n^s | (x,n^s)_s = d^s_j . Bibak et\nal. gave a formula using Ramanujan sums for the number of solutions of the\nabove congruence equation with some gcd restrictions on x_i. We generalize\ntheir result with generalized gcd restrictions on x_i by proving that for the\nabove linear congruence, the number of solutions is\n
frac1n^s
sum
limits_d|nc_d,s(b)
prod
limits_j=1
^tau(n)
left(c_
fracnd_j,s(
fracn^sd^s)
right)^g_j\nwhere g_j = | x₁,\…, x_k \∩ \C_j,s| for j=1,\…\n\τ(n) and c_d,s denote the generalized ramanujan sum defined by E.\nCohen.\n

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