Prime number lists

Question

"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.

How did you try out "every combination of numbers"? Are you implying that you have found $p$ and $n$ that result in $p, p+n, p+2n, p+3n, p+4n$ as prime numbers? Can you put in some more detail of how you found $p$ and $n$? — an4s, Jul 07 '20 at 16:26
you found $n$ must be a multiple of $2$; must $n$ be a multiple of $3$? — J. W. Tanner, Jul 07 '20 at 16:29
Okay I haven't tried literally every combination I just wanted to see if there was a better way of doing this then brute force. — user287133, Jul 07 '20 at 16:30
an4s I simply went through every k starting from 1 and found n=6 and p=5 to work but it does not work for p+5n since 35 is not prime — user287133, Jul 07 '20 at 16:35
Consider changing the title to "Prime Numbers in Arithmetic Progression" — J. W. Tanner, Jul 07 '20 at 19:32

J. W. Tanner · Answer 1 · 2020-07-07T16:47:43.790

2

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 $p=7$.

edited Jul 07 '20 at 16:47

answered Jul 07 '20 at 16:40

J. W. Tanner

60,406

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 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
1

Yes, $n$ must be a multiple of $30$, and $p$ cannot be $2, 3, $ or $5$; cf. this Wikipedia article – 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

Prime number lists

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