Mersenne Primes, Pt. 1

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 about using a base other than $2$ for generating primes. How many prime numbers are of the form $a^b-1$ , with positive integers $a,b$ such that $a>2$ and $b>1$ ?

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Exactly

2

Exactly

1

Not known (depends on number of Mersenne primes) Infinite Exactly

3

Finite, but more than

3

None

1 solution

X X
May 15, 2018

$a^b-1=(a-1)(a^{b-1}+a^{b-2}+a^{b-3}+\cdots+a^2+a^1+1)$ When $a>2,a-1>1$ ,when $b>1,a^{b-1}>1$ , multiplying two positive integers larger than one will never get a prime.

Mersenne Primes, Pt. 1

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