New Solvability Conditions for Congruence

K. Bibak et al. [arXiv:1503.01806v1 [math.NT], March 5 2015] proved that congruence ax ≡ b (mod n) has a solution x0 with t = gcd(x0, n) if and only if gcd ( a, n t ) = gcd ( b t , n t ) thereby generalizing the result for t = 1 proved by B. Alomair et al. [J. Math. Cryptol. 4 (2010), 121–148] and O. Grošek et al. [ibid. 7 (2013), 217–224]. We show that this generalized result for arbitrary t follows from that for t = 1 proved in the later papers. Then we shall analyze this result from the point of view of a weaker condition that gcd ( a, n t ) only divides gcd ( b t , n t ) . We prove that given integers a, b, n ≥ 1 and t ≥ 1, congruence ax ≡ b (mod n) has a solution x0 with t dividing gcd(x0, n) if and only if gcd ( a, n t ) divides gcd ( b t , n t ) . Ga u ß revolutionized the number theory with the idea of the congruence in his D.A. He introduced congruence in the very first article of D.A., and the following basic result on the solvability of linear congruence1 ax ≡ b (mod n) (1) which belongs to standard requisites of elementary number theory can be found in Arts. 29, 30 of D.A. (cf. [4]): 1 If a, b, n ∈ Z, and gcd(a, n) = d, then the congruence (1) is solvable if and only if d|b. c © 2015 Mathematical Institute, Slovak Academy of Sciences. 2010 Mathemat i c s Sub j e c t C l a s s i f i c a t i on: Primary 11A07; Secondary 11D04, 11D45, 11A25, 11B50.