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
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$\begingroup$ no it isn t a fixed number we can change it , it depends to the value of and b $\endgroup$ – contestant IMO 2020 Sep 11 '18 at 12:28
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$\begingroup$ @AbderrahmaneDriouch That is not what your question says. In your question, $s$ cannot depend on $a$ or $b$. $\endgroup$ – Batominovski Sep 11 '18 at 12:29
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$\begingroup$ This question does not make sense, please edit for clarity. $\endgroup$ – lulu Sep 11 '18 at 12:30
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1$\begingroup$ Voting to close the question as it is unclear what you are asking. If you can, please edit your post for clarity. $\endgroup$ – lulu Sep 11 '18 at 12:33
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2$\begingroup$ 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. $\endgroup$ – asdf Sep 11 '18 at 12:33
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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 $s$.
answered Sep 11 '18 at 12:38
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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$
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$\begingroup$ Thank you for the correct addition; I hope you don't mind that I've edited it into my answer. $\endgroup$ – Mees de Vries Sep 11 '18 at 13:29
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1$\begingroup$ @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. $\endgroup$ – mathguy Sep 11 '18 at 13:34
