This question recently came up to me in a project and is not taken from a textbook. I would like to know if any characterization of such primes is known from literature. They are seemingly rare but do exist. The first five examples are: $3$, $7$, $31$, $127$, $8191$. Note that these are all congruent to $3$ modulo $4$. The first four are also exponents of Mersenne primes, although $8191$ is not. I have also noted that they all satisfy $p^2 \equiv -1 \pmod{\operatorname{rad}((p+1)(p^2+1))}$, where $\operatorname{rad}(k)$ denotes the product of the prime divisors of an integer $k$. In particular this would imply that $\operatorname{rad}(p+1) \mid \operatorname{rad}(p^2+1)$ as well, and that $-1$ is a quadratic residue modulo each of the prime divisors of $(p + 1)(p^2 + 1) = (p^4-1)/(p-1)$; hence each prime divisor of $(p+1)$ must be congruent to $1$ modulo $4$. Many thanks for any insights.
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Suppose a prime $q$ divides $p+1$. We are told that $q$ divides $p-1$, so $q=2$. – André Nicolas May 20 '14 at 02:12
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1Well, the only prime that can divide both $p+1$ and $p-1$ is $2$, so the set of primes you're asking about is the set of Mersenne Primes. – Nishant May 20 '14 at 02:13
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@Nishant: You might as well answer. – André Nicolas May 20 '14 at 02:17
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Thanks a lot guys. Feel silly now :) – May 20 '14 at 02:25