New solvability conditions for congruence $ax\equiv b\pmod n$

Rok, strany: 2015, 93 - 99

K. Bibak \emph{et al}. [arXiv:1503.01806v1 [math.NT], March 5 2015] proved that congruence $ax\equiv b\pmod{n}$ 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 \emph{et al}. [J. Math. Cryptol. \textbf{4} (2010), 121–148] and O. Gro\v{s}ek \emph{et al}. [\emph{ibid}. \textbf{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\vadjust{\vskip0.75pt} 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 $ax\equiv b\pmod{n}$ has a solution $x₀$ with $t$ dividing $\gcd(x₀,n)$ if and only if $\gcd(a,((n) / (t)))$ divides $\gcd(((b) / (t)),((n) / (t)))$.

ISO 690:

Porubský, Š. 2015. New solvability conditions for congruence $ax\equiv b\pmod n$. In Tatra Mountains Mathematical Publications, vol. 64, no.3, pp. 93-99. 1210-3195.

APA:

Porubský, Š. (2015). New solvability conditions for congruence $ax\equiv b\pmod n$. Tatra Mountains Mathematical Publications, 64(3), 93-99. 1210-3195.