A rationality condition for the existence of odd perfect numbers

Simon Davis

doi:10.1155/s0161171203108277

Abstract

A rationality condition for the existence of odd perfect numbers is used to derive an upper bound for the density of odd integers such that σ(N) could be equal to 2N, where N belongs to a fixed interval with a lower limit greater than 10300. The rationality of the square root expression consisting of a product of repunits multiplied by twice the base of one of the repunits depends on the characteristics of the prime divisors, and it is shown that the arithmetic primitive factors of the repunits with different prime bases can be equal only when the exponents are different, with possible exceptions derived from solutions of a prime equation. This equation is one example of a more general prime equation, , and the demonstration of the nonexistence of solutions when h ≥ 2 requires the proof of a special case of Catalan′s conjecture. General theorems on the nonexistence of prime divisors satisfying the rationality condition and odd perfect numbers N subject to a condition on the repunits in factorization of σ(N) are proven.

Highlights

The algorithm for demonstrating the nonexistence of odd perfect numbers with fewer than nine different prime divisors requires the expansion of the ratio σ (N)/N and strict inequalities imposed on the sums of powers of the reciprocal of each prime divisor [20, 55]
Since the integers in (4.6) and (4.8) did not satisfy the condition σ (N)/N = 2 and the rationality condition is satisfied by selected sets of primes only, the results provide support for the nonexistence of odd perfect numbers for large categories of prime divisors and exponents, which will be established in Theorem 7.1
The rationality condition provides an analytic method for investigating the existence of odd perfect numbers, as it would be sufficient to demonstrate that there is an unmatched prime divisor in the product

Summary

A RATIONALITY CONDITION FOR THE EXISTENCE OF ODD PERFECT NUMBERS

A rationality condition for the existence of odd perfect numbers is used to derive an upper bound for the density of odd integers such that σ (N) could be equal to 2N, where N belongs to a fixed interval with a lower limit greater than 10300. The rationality of the square root expression consisting of a product of repunits multiplied by twice the base of one of the repunits depends on the characteristics of the prime divisors, and it is shown that the arithmetic primitive factors of the repunits with different prime bases can be equal only when the exponents are different, with possible exceptions derived from solutions of a prime equation This equation is one example of a more general prime equation, (qjn − 1)/(qin − 1) = ph, and the demonstration of the nonexistence of solutions when h ≥ 2 requires the proof of a special case of Catalan’s conjecture.

Introduction

Conclusion