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As I was playing around with Mersenne numbers, and discovered the notion of Wagstaff prime going off Wikipedia, I started considering the sequence, for a given $odd$ prime number $p$, defined as follows: $2^{p}-1, \frac{2^{p}+1}{3}, \frac{2^{p}+3}{5}, \cdots \frac{2^{p}+p}{p+2}$.

It seems that the first term of this sequence is a prime if and only if all other terms but one are positive integers.

Is this true?

Thanks in advance.

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    $\begingroup$ Voted to close as off-topic -- counterexamples are much too easily found for this being a 'research-level' question. $\endgroup$
    – Stefan Kohl ♦
    Commented Nov 10, 2013 at 22:59

1 Answer 1

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$2^{19}-1$ is prime, but neither $(2^{19}+3)/5$ nor $(2^{19}+9)/11$ is an integer.

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    $\begingroup$ 2^17-1 is an even smaller counterexample. $\endgroup$
    – Stefan Kohl ♦
    Commented Nov 10, 2013 at 22:55
  • $\begingroup$ @Stefan, I'm not surprised. I didn't check systematically, I just guessed that 19 would be big enough to fail and small enough to test with little effort. $\endgroup$
    – Gerry Myerson
    Commented Nov 10, 2013 at 22:57

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