On Prime-Perfect Numbers

Abstract

We prove that the Diophantine equation p r 1 1 · · · p r k k =`r 1 +···+r k r 1 ,...,r k´h as only finitely many positive integer solutions k, p 1 , . . . , p k , r 1 , . . . , r k , where p 1 , . . . , p k are distinct primes. If a positive integer n has prime factorization p r 1 1 · · · p r k k , then`r 1 +···+r k r 1 ,...,r k´r epresents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equation are designated as prime-perfect numbers.