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

Statement. Among the even numbersMathworldPlanetmath, only powers of two 2x (with x being a nonnegative integer) can be 2-superperfect numbers (, and then if and only if 2x+1-1 is a Mersenne primeMathworldPlanetmath. (The default multiplier m=2 is tacitly assumed from this point forward).

Proof. The only divisorsMathworldPlanetmathPlanetmath of n=2x are smaller powers of 2 and itself, 1,2,,2x-1,2x. Therefore, the first iteration of the sum of divisors function is


If 2n-1 is prime, that means its only other divisor is 1, and thus for the second iteration σ(2n-1)=2n, and is thus a 2-superperfect number. But if 2n-1 is composite then it is clear that σ(2n-1)>2n by at least 2. So, for example, σ(8)=15 and σ2(8)=24, so 8 is not 2-superperfect. One more example: σ(16)=31 and since 31 is prime, σ2(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 functionMathworldPlanetmath, it follows that if n=2xp then σ(n)=σ(2x)σ(p). So, if, say, p=3, it is clear that (2x+3-4)>2x+13, and that on the second iteration this value that already exceeded twice the original value will be even greater. For example, 12=223, and σ(12)=25-4 which is greater than 233 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
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