A rationality condition for the existence of odd perfect numbers (Q1811904)

Summary: 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 \(\sigma(N)\) could be equal to \(2N\), where \(N\) belongs to a fixed interval with a lower limit greater than \(10^{300}\). 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, \((q_{j}^{n}-1)/{(q_{i}^{n}-1)}=p^h\), and the demonstration of the nonexistence of solutions when \(h\geq 2\) 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 \(\sigma(N)\) are proven.