Proof that there are infinitely many $k$'s such that $a + k$ and $b + k$ are coprime

Question

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.

score 2 · Accepted Answer · answered Jan 16 '19 at 10:32

2

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$$\gcd(a+k,a-b)=\gcd\bigl(a+k,(a+k)-(b+k)\bigr)=\gcd(a+k,b+k).$$

answered Jan 16 '19 at 10:32

José Carlos Santos

427,504

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. – DanielWainfleet Jan 16 '19 at 13:51
@DanielWainfleet That's another reason, although I must say that I find that justification better than mine. – José Carlos Santos Jan 16 '19 at 13:53
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 – DanielWainfleet Jan 16 '19 at 21:12
Please strive not to add more (dupe) answers to dupes of FAQs. – Bill Dubuque Mar 10 '22 at 20:17
@BillDubuque Really? For an answer posted more than three years ago? – José Carlos Santos Mar 10 '22 at 20:18
@José Yes, really, posting (dupe) answers to obvious FAQs have always been frowned upon (except in early "population" days). Did you really think that such a common exercise hadn't been asked and answered in the prior nine years? It has - many times. – Bill Dubuque Mar 10 '22 at 20:20

Proof that there are infinitely many $k$'s such that $a + k$ and $b + k$ are coprime

1 Answers1