On some variations of perfect and multiperfect numbers

A positive integer is said to be perfect if the sum of its divisors is twice the number. This paper is a partial exposition of the articles On Multiply Perfect Numbers with a Special Property by Carl Pomerance (1975), Variations on Euclid's Formula for Perfect Numbers by Farideh Firoozbakt (201...

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Bibliographic Details
Main Authors: Quintana, Noel Enrico, Sancho, Miguel Antonio
Format: text
Language:English
Published: Animo Repository 2010
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Online Access:https://animorepository.dlsu.edu.ph/etd_bachelors/5334
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Institution: De La Salle University
Language: English
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Summary:A positive integer is said to be perfect if the sum of its divisors is twice the number. This paper is a partial exposition of the articles On Multiply Perfect Numbers with a Special Property by Carl Pomerance (1975), Variations on Euclid's Formula for Perfect Numbers by Farideh Firoozbakt (2010), and Iterating the Sum-of-Divisors Function by Graeme Cohen and Herman te Riele. The paper deals with different variations of perfect numbers. The first major result determined solutions to a problem involving the sum-of-divisors function satisfying certain conditions. The results showed that all solutions are multiperfect numbers, a generalization of perfect numbers in which the sum of the divisors is a positive integral multiple of the number. The second article dealt with generalizing the concept of multiperfect numbers by iterating the sum-of-divisors function σ, giving rise to what we call (m,k)-perfect numbers. One of the questions that was attempted to e answered was Are all numbers (m,k)-perfect? It was shown that numbers up to 1000 are (m,k)-perfect. The third article gave solutions to various variations of Euclid's equation of the form σ(x), = kx + f(x), where k is an integer larger than 1 and f is an arithmetic function of x.