Solving the odd perfect number problem: Some old and new approaches
Date of Publication
2008
Document Type
Master's Thesis
Degree Name
Master of Science in Mathematics
College
College of Science
Department/Unit
Mathematics and Statistics
Thesis Adviser
Severino V. Gervacio
Defense Panel Chair
Leonor A. Ruivivar
Defense Panel Member
Sonia Y. Tan
Fidel R. Nemenzo
Abstract/Summary
A perfect number is a positive integer} N such that the sum of all the positive divisors of N equals 2N, denoted by Sigma(N) = 2N. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems of number theory. This thesis presents some of the old as well as new approaches to solving the OPN Problem. In particular, a conjecture predicting an injective and surjective mapping X = Sigma(p^k) / p^k Y = Sigma(m^2) / m^2 between OPNs N = (p^k)(m^2) (with Euler factor p^k) and rational points on the hyperbolic arc XY = 2 with 1 < X < 1.25 < 1.6 < Y < 2 and 2.85 < X+Y < 3, is disproved. Various results on the abundancy index and solitary numbers are used in the disproof. Numerical evidence against the said conjecture will likewise be discussed. We will show that if an OPN N has the form above, then p^k < ( 2 / 3)(m^2) follows from [15]. We will also attempt to prove a conjectured improvement of this last result to p^k < m by observing that Sigma(p^k) / m is not equal to1 and Sigma(p^k) / m is not equal to Sigma(m) / p^k in all cases. Lastly, we also prove the following generalization: If N = Prod_{i=1}^{r}{p_i^{ \ alpha_i}} is the canonical factorization of an OPN N, then Sigma( {p_i} )^{ \ alpha_i} ) < = (2 / 3)[N / {p_i}^{ \ alpha_i})] for all i. This gives rise to the inequality N^(2-r) < = (1 / 3)(2 / 3)^(r-1), which is true for all r, where r= omega(N) is the number of distinct prime factors of N.
Abstract Format
html
Format
Electronic
Accession Number
CDTG004463
Shelf Location
Archives, The Learning Commons, 12F Henry Sy Sr. Hall
Physical Description
1 computer optical disc ; 4 3/4 in.
Keywords
Number theory; Perfect numbers
Recommended Citation
Dris, J. B. (2008). Solving the odd perfect number problem: Some old and new approaches. Retrieved from https://animorepository.dlsu.edu.ph/etd_masteral/3724
