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An even integer $n>0$ is a perfect number if and only ifThis forces $p$ to be a prime number, as shown under Mersenne number. It does not say that every prime exponent works, nor classify possible odd integers.^[ref]

For the forward direction, put $q=2^p-1$ and use coprimality of $q$ and $2^{p-1}$. Multiplicativity of the sum-of-divisors function and the finite geometric series giveThe prime number hypothesis makes $\sigma (q)=q+1$; replacing it with a composite number changes this factor.

For a short converse outline, write $n=2^{a}m$ with $a\ge 1$ and $m$ an odd integer. The condition $\sigma (n)=2n$ givesHere $t$ is a positive integer and a proper divisor of $m$. If $t>1$, the distinct divisors $1,t,m$ alone exceed the required sum $m+t$. Thus $t=1$ and $\sigma (m)=m+1$, forcing $m$ to be prime and $m=2^{a+1}-1$.

Marin Mersenne was a mathematician and scientific correspondent whose network helped Pierre de Fermat, René Descartes and other researchers exchange questions and results. He studied number theory and the acoustics of vibrations of mechanical strings.^[ref] A Mersenne number has the form $2^p-1$; for it to be a Mersenne prime, $p$ must be a prime number, although that condition is not sufficient: $2^{11}-1=23\cdot 89$. For an ideal stretched mechanical string, the fundamental frequency is $f=(2L{)}^{-1}\sqrt{T/\mu }$, relating the audible pitch to the length $L$, tension $T$ and mass per unit length $\mu$.

A Mersenne number has the form $M_k=2^k-1$ for a positive integer $k$. Its binary representation consists of $k$ ones. For example, $M_4=15$ is composite, while $M_5=31$ is prime. If $k=ab$ with $a,b>1$, the finite geometric series factorizationhas two factors greater than one. Consequently a Mersenne prime requires a prime exponent, but $M_{11}=23\cdot 89$ shows that this condition alone is insufficient.

A Mersenne prime is a Mersenne number that is a prime number. It is the prime factor required by the Euclid-Euler theorem; $3,7,31,127$ generate $6,28,496,8128$. The family is named after Marin Mersenne.