Recently, I've learned how to solve diophantine linear equations (i.e. of form $ax + by = c$) using euclidean algorithm. I am wondering about how big $x$ and $y$ can get in terms of $a, b, c$. Are there any estimates?
Estimate for $x, y$ values in linear diophantine equation
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number-theory
diophantine-equations
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0If the equation has one solution, it has infnitely many, so there's no bound on $x$ and $y$. – 2017-02-07
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0Oh, sorry. I meant the $x, y$ after performing euclidean algorithm. I know that there are infinitely many solutions, but my problem is that I want to write a program which would solve linear diophantine equation and I don't know if $x, y$ would overflow, because I have only 64-bit variables available. – 2017-02-07
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1OK. Let $d=\gcd(a,b,c)$. Then Euclid will give you a solution of $ax+by=d$ with $|x|<|b|$ and $|y|<|a|$. Then multiply by $|c/d|$. – 2017-02-07