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.
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.