1
$\begingroup$

I can't solve this, can you help me please?
What's the minimum value of $k\in\mathbb{Z}^+$ such that $a+nk$ is prime with $n\in\{0,1,\dots,5\}$.

1 Answers 1

2

I think OEIS A006560 aka CPAP-k series might be of help.

First and smallest occurrence of $n$, $n >= 1$, consecutive primes in arithmetic progression:

$a(1) = 2: (2)\\ a(2) = 2: (2, 3)\\ a(3) = 3: (3, 5, 7)\\ a(4) = 251: (251, 257, 263, 269)\\ a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139)\\ a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961)$

Since you are asking for minimum $k$ but not $n$. The common differences of first and smallest AP of $n \ge 1$ consecutive primes:

${0, 1, 2, 6, 30, 30, \ge 210, \ge 210, \ge 210, \ge 210, \ge 2310, ...}$

So minimum $k$ for PAP-6 is $30$.

Also this page about PAP is helpful.

  • 0
    How can I prove it?, I need to find it (I know it is 30 now) and prove it. – 2017-01-21
  • 0
    Most of the prime related series/problems are based on observations and then conjectured. Such series are computer generated by using certain algorithms. – 2017-01-21
  • 0
    I tried searching and came across [Green-Tao theorem](https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem) but it was beyond my scope of understanding. The theorem itself is based on some other conjectures and theorems. Maybe it might help you. – 2017-01-21