Give two positive integers $a$ and $b$, prove that there is an infinity of integers $s$ so that $a+s$ and $b+s$ are co-prime. I have an idea that is to use the fact that the GCD of (a,b) is the same as (a-b,b)... But it leads me nowhere
2026-03-30 00:22:58.1774830178
Show that there is an infinity of integers $s$ so that $a+s$ and $b+s$ are co-prime.
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Note that $\gcd(a+s,b+s) = \gcd((a+s)-(b+s),b+s) = \gcd(a-b,b+s)$. It suffices to show that there are infinitely many $s$ such that $b+s$ is coprime to $a-b$. For instance, take $s>a$ such that $b+s$ is prime. By $b+s > b+a > a-b > -s - b$ we know that $b+s$ and $a-b$ are coprime.