# Prove that $\gcd(a+s,b+s)=1$ for infinitely many numbers $s$

Prove that for every natural numbers $a$ and $b$, there are infinitely many numbers $s$, such $\gcd(a+s,b+s)=1$ and $a\neq b$

I tried to use Bezout's theorem but I can't get to the result

• no it isn t a fixed number we can change it , it depends to the value of and b – contestant IMO 2020 Sep 11 '18 at 12:28
• @AbderrahmaneDriouch That is not what your question says. In your question, $s$ cannot depend on $a$ or $b$. – Batominovski Sep 11 '18 at 12:29
• This question does not make sense, please edit for clarity. – lulu Sep 11 '18 at 12:30
• Voting to close the question as it is unclear what you are asking. If you can, please edit your post for clarity. – lulu Sep 11 '18 at 12:33
• I edited the question - is this what you wanted to ask? And don't demand solutions from people, no one here is obligated to solve YOUR problems for YOU. – asdf Sep 11 '18 at 12:33

As rightfully pointed out in the other answer, the statement trivially does not hold when $a = b$. Thus we must make the assumption that $a \neq b$. Then, my original answer holds:

Without loss of generality, $a > b$. A property of the greatest common divisor tells us that $$\gcd(a + s, b + s) = \gcd((a + s) - (b + s), b + s) = \gcd(a - b, b + s).$$ Hence, if $p$ is any prime that does not divide $a - b$, we can choose $s = p - b$, and then $$\gcd(a + s, b + s) = \gcd(a - b, p) = 1.$$ Since there are infinitely many primes, this shows that there are infinitely many such $s$.

As stated the claim is obviously false: $a$ and $b$ may be equal, in which case the GCD is $|a+s|$ (assuming you meant integer $s$, otherwise the problem doesn't make sense), and that is equal to $1$ for exactly two values of $s$.

So you must also have the assumption that $a \ne b$. Let's assume $a < b$.

There are infinitely many primes greater than $b$. For any such prime $p$, let $s = p - b$. Then $\gcd(a+s,b+s)=\gcd(a+p-b, p) = 1$, because $p$ is prime and $1 < a+p-b < p$

• thanks......... – contestant IMO 2020 Sep 11 '18 at 13:28
• Thank you for the correct addition; I hope you don't mind that I've edited it into my answer. – Mees de Vries Sep 11 '18 at 13:29
• @MeesdeVries - Of course, especially since you also specifically referenced the other answer, you didn't simply change yours. That is the proper way to deal with such situations IMO. I wrote my answer and posted it without seeing yours. Then I saw yours and noticed it is very similar; but I left mine up, since I don't see the need to go through $a-b$. My thought was to go directly to $b+s$ being a prime, so the GCD should automatically be $1$, with essentially no other justification needed. – mathguy Sep 11 '18 at 13:34