Mersenne Primes, Pt. 2

Number Theory Level 3

A Mersenne prime is any prime of the form $2^p-1$ , where $p>1$ is a positive integer. As of May 2018, there are only 50 Mersenne primes known (the largest being $2^{77,232,917}-1$ with $23,249,425$ digits). It is still an open question as to whether there are an infinite number of Mersenne primes.

One may wonder what happens when one generalizes the Mersenne prime. How many prime numbers are of the form $a^p-b^p$ , with positive integers $a,b,p$ such that $a>2$ , $b>1$ , and $p>1$ ?

If you enjoyed this problem, you might want to check out this one .

Finite, but more than

3

Exactly

2

Exactly

1

Exactly

3

Not known (depends on number of Mersenne primes) Infinite

1 solution

Mark Hennings
May 15, 2018

With $a = b+1$ and $p=2$ we have $a^p - b^p = 2b+1$ for all $b > 1$ . Thus all odd numbers except $3$ can be obtained by this formula. Since there are infinitely many primes, and all but one of these is odd, we can obtain infinitely many (all but two of them) primes by this formula.

Mersenne Primes, Pt. 2

1 solution

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