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Given integer constants $a,b,d$ ($a,b$ are coprime) and an integer variable $x$, is there a way to easily find for which $x$ does $gcd(a^2-db^2,a-bx)=a^2-db^2$ without substituting the values of $x$ individually and calculating the gcd? This is a continuation oy my gcd studies following the link Relation between two gcds.

Thanks

John Bull
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  • As in prior\ question: let $,e = a^2-db^2.,$ Then $,(e,b) = (a^2,b)=1,$ by $,(a,b)=1,,$ so $,\color{#c00}{b^{-1}}$ exists $\bmod e.,$

    $,(e,a-bx) = e\iff e\mid a-bx\iff x\equiv a\color{#c00}{b^{-1}}\pmod{e}\ \ $

     – Bill Dubuque May 16 '23 at 17:02

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