proof of the correspondence between even 2-superperfect numbers and Mersenne primes
Statement. Among the even numbers, only powers of two (with being a nonnegative integer) can be 2-superperfect numbers (http://planetmath.org/SuperperfectNumber), and then if and only if is a Mersenne prime. (The default multiplier is tacitly assumed from this point forward).
If is prime, that means its only other divisor is 1, and thus for the second iteration , and is thus a 2-superperfect number. But if is composite then it is clear that by at least 2. So, for example, and , so 8 is not 2-superperfect. One more example: and since 31 is prime, .
Now it only remains to prove that no other even number 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 . Since the sum of divisors function is a multiplicative function, it follows that if then . So, if, say, , it is clear that , and that on the second iteration this value that already exceeded twice the original value will be even greater. For example, , and which is greater than by 4. With any larger the excess will be much greater.
|Title||proof of the correspondence between even 2-superperfect numbers and Mersenne primes|
|Date of creation||2013-03-22 17:03:48|
|Last modified on||2013-03-22 17:03:48|
|Last modified by||PrimeFan (13766)|