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"Find the smallest possible integer n with the property that there exists a prime $p$ such that the $6$ numbers: $p, p+n, p+2n, p+3n, p+4n, p+5n$ are all prime numbers."

Okay, so I have tried what I thought to be every combination of numbers and cannot figure out what works for the last number $p+5n$; I understand that $n$ must be even since if $n$ is odd then there will always be at least $2$ even numbers. which would be a contradiction of all numbers being prime. Any hints would be greatly appreciated on solving this without brute force.

J. W. Tanner
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user287133
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1 Answers1

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Hint:

If $n$ is not a multiple of $2$, then $p+n$ or $p+2n$ is.

If $n$ is not a multiple of $3$, then $p+n$ or $p+2n$ or $p+3n$ is.

If $n$ is not a multiple of $5$, then $p+n$ or $p+2n$ or $p+3n$ or $p+4n$ or $p+5n$ is.

With that information, try $p=7$.

J. W. Tanner
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  • I am confused if n is not a multiple of 5, how is the rest a multiple of 5 without knowing what p is? – user287133 Jul 07 '20 at 17:14
  • The answers to [this question](https://math.stackexchange.com/questions/882609/first-index-of-number-in-that-arithmetic-progression-which-is-a-multiple-of-the) may help you realize that *one* of $p+n, p+2n, p+3n, p+4n$, or $p+5n$ will be a multiple of $5$ if $n$ is not – J. W. Tanner Jul 07 '20 at 17:20
  • Okay the smallest combination I have realized is p=7 and n=30, is this the smallest? – user287133 Jul 07 '20 at 17:28
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    Yes, $n$ must be a multiple of $30$, and $p$ cannot be $2, 3, $ or $5$; cf. [this Wikipedia article](https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression) – J. W. Tanner Jul 07 '20 at 17:29
  • why is it that p cannot be 2,3 or 5 sorry I am confused by that part. – user287133 Jul 07 '20 at 17:37
  • if $p$ is $2,3, $ or $5$, then $p+30$ is a multiple of $2, 3, $ or $5$, respectively, and therefore not prime – J. W. Tanner Jul 07 '20 at 17:38