For a problem in group Theory I need some information about the Mersenne primes:

Let $p = 2^a - 1 > 7$ be a Mersenne prime and so $a$ is an odd prime. Is it true that $p^2 + 1$ is square free. i.e. if there exists a prime number $q$ such that $q$ divides $(p^2+1)/2$, then $q^2$ does not didivde $(p^2+1)/2$?

Also is there any result about the number of distinct prime divisors of $p^2+1$ by these asumptions?

Many thanks for your help

BHZ

For a problem in group Theory I need some information about the Mersenne primes:

Let $p=2^a-1>7$ be a Mersenne prime and so $a$ is an odd prime. Is it true that $p^2+1$ is square free. i.e. if there exists a prime number $q$ such that $q$ divides $(p^2+1)/2$, then $q^2$ does not didivde $(p^2+1)/2$?

Also is there any result about the number of distinct prime divisors of $p^2+1$ by these assumptions?

Many thanks for your help

BHZ

For a problem in group Theory I need some information about the Mersenne primes:

Let $p=2^a-1>7$ be a Mersenne prime and so $a$ is an odd prime. Is it true that $p^2+1$ is square free. i.e. if there exists a prime number $q$ such that $q$ divides $(p^2+1)/2$, then $q^2$ does not didivde $(p^2+1)/2$?

Also is there any result about the number of distinct prime divisors of $p^2+1$ by these asumptions?

Many thanks for your help

BHZ

For a problem in group Theory I need some information about the Mersenne primes:

Let $p = 2^a - 1 > 7$ be a Mersenne prime and so $a$ is an odd prime. Is it true that $p^2 + 1$ is square free. i.e. if there exists a prime number $q$ such that $q$ divides $(p^2+1)/2$, then $q^2$ does not didivde $(p^2+1)/2$?

Also is there any result about the number of distinct prime divisors of $p^2+1$ by these asumptions?

Many thanks for your help

BHZ