ON THE FORM OF PERFECT AND MULTIPERFECT NUMBERS

摘要

Multiperfect numbers of (integer) multiplicity k, also known as k-perfect numbers, are positive integers n satisfying the property σ(n) = kn, where σ (n) is the "sum of divisors" function: σ(n) =∑_(d|n)d. In the special case where k = 2, then the k-perfect number n is said to be perfect. Both multiperfect and the better known perfect numbers have been the source of great interest and mystery over the past two millennia. Are there infinitely many perfect or multiperfect numbers? Does there exist an odd perfect or multiperfect number? Nobody knows the answer to these questions, and progress toward an answer to either of them has been minimal. In fact, there is very little in the research literature focusing on the form of multiperfect numbers.