# proof of the correspondence between even 2-superperfect numbers and Mersenne primes

Statement. Among the even numbers^{}, only powers of two ${2}^{x}$ (with $x$ being a nonnegative integer) can be 2-superperfect numbers (http://planetmath.org/SuperperfectNumber), and then if and only if ${2}^{x+1}-1$ is a Mersenne prime^{}. (The default multiplier $m=2$ is tacitly assumed from this point forward).

Proof. The only divisors^{} of $n={2}^{x}$ are smaller powers of 2 and itself, $1,2,\mathrm{\dots},{2}^{x-1},{2}^{x}$. Therefore, the first iteration of the sum of divisors function is

$$\sigma (n)=\sum _{i=0}^{x}{2}^{i}={2}^{x+1}-1=2n-1.$$ |

If $2n-1$ is prime, that means its only other divisor is 1, and thus for the second iteration $\sigma (2n-1)=2n$, and is thus a 2-superperfect number. But if $2n-1$ is composite then it is clear that $\sigma (2n-1)>2n$ by at least 2. So, for example, $\sigma (8)=15$ and ${\sigma}^{2}(8)=24$, so 8 is not 2-superperfect. One more example: $\sigma (16)=31$ and since 31 is prime, ${\sigma}^{2}(16)=32$.

Now it only remains to prove that no other even number $n$ can be 2-superperfect. Any other even number can of course still be divisible by one or more powers of two, but it also must be divisible by some odd prime $p>2$. Since the sum of divisors function is a multiplicative function^{}, it follows that if $n={2}^{x}p$ then $\sigma (n)=\sigma ({2}^{x})\sigma (p)$. So, if, say, $p=3$, it is clear that $({2}^{x+3}-4)>{2}^{x+1}3$, and that on the second iteration this value that already exceeded twice the original value will be even greater. For example, $12={2}^{2}3$, and $\sigma (12)={2}^{5}-4$ which is greater than ${2}^{3}3$ by 4. With any larger $p$ the excess will be much greater.

Title | proof of the correspondence between even 2-superperfect numbers and Mersenne primes |
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Canonical name | ProofOfTheCorrespondenceBetweenEven2superperfectNumbersAndMersennePrimes |

Date of creation | 2013-03-22 17:03:48 |

Last modified on | 2013-03-22 17:03:48 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 5 |

Author | PrimeFan (13766) |

Entry type | Proof |

Classification | msc 11A25 |