Which is the largest integer $n<1000$ so that $n$, $n+2$ and $n+4$ are primes?
I have tried to solve this problem but have not reached an argument worth
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2Think about divisibility by $3$. – Gerry Myerson Feb 25 '13 at 05:25
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I'm pretty sure that the letter y between $n+2$ and $n+4$ means the same thing as "and" in English, so I edited that in. – Will Jagy Feb 25 '13 at 05:33
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n = 3, because for any larger n, one of them is divisible by 3.
marty cohen
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@ErickWong: It makes no difference. They were asking for the largest integer. Even if we consider largeness in terms of absolute value, and even if we consider negative primes, the answer is still $3$. – Cameron Buie Feb 25 '13 at 05:58
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2@ErickWong: /facepalm/ Huh. Yep. $7$ is bigger than $3$. I give up for the night. – Cameron Buie Feb 25 '13 at 06:11
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Observe the following. If $n$ is prime, let it be $1$ mod $3$. Then $n+2$ will be divisible by $3$. If it is $2$ mod $3$, $n+4$ will be divisible by $3$. Therefore the only solution is if $n=3$. i.e. $3,5,7$ is the sequence you are after.
Gautam Shenoy
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