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I need to show that for any $a, b \in \mathbb{Z}^+$ with $a \neq b$ there are infinitely many $k \in \mathbb{Z}$ such that $a + k$ and $b + k$ are relatively prime to each other.

I came up with a proof that uses the fact that there are infinitely many primes and that we can always choose $k$ such that $a + k$ is prime and therefor $b + k$ is relatively prime to $a + k$ (assuming that $a > b$).

But I was given the hint $\gcd(x, y) = \gcd(x, y - zx)$ which I don't use in the proof that I came up with. Hence my question is, if there is a way to show this fact by only using this hint and some other basic facts about the gcd.

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Assume that $a>b$. There are infinitely many numbers $m>a$ such that $\gcd(m,a-b)=1$ (since $a-b$ only has finitely many prime factors). In other words, there are infinitely many $k$'s such that $\gcd(a+k,a-b)=1$. But$$\gcd(a+k,a-b)=\gcd\bigl(a+k,(a+k)-(b+k)\bigr)=\gcd(a+k,b+k).$$

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  • $\begingroup$ The reason there are infinitely many $ m>a$ co-prime to $a-b$ is that $\{1+n|a-b|: n\in \Bbb N\}$ is infinite. $\endgroup$ – DanielWainfleet Jan 16 at 13:51
  • $\begingroup$ @DanielWainfleet That's another reason, although I must say that I find that justification better than mine. $\endgroup$ – José Carlos Santos Jan 16 at 13:53
  • $\begingroup$ The only reason I mentioned it is that the proposer wanted to see it done without using the fact that there are infinitely many primes....+1 $\endgroup$ – DanielWainfleet Jan 16 at 21:12

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