Infinitude Condition for Mersenne and Sophie Germain Primes

This paper explores both Mersenne primes of the form where, is a prime. By extension, the paper also explores Perfect numbers and Sophie primes. Botgh these special primes have a special relationship for “perfectitude”. An insight into these numbers is explored using novel methods that involve the trigonometric functions with integer factorable arguments. Rational functions play a part in the behavior of many functions including regular primes, Mersenne Primes, and Perfect numbers. The paper first determines relationships for primes, and then procedes to show how Perfect number relations can be derived from trigonometric relations. The relationships of trigomentric functions involving the sum of divisors, provide a novel approach to prove that that the analytic structure of cot(x), when split into Mersenne and non-Mersenne classes through the Bernoulli framework, forces a coupling between the two infinite subsets of integers and the contradiction (negative ratio despite all positive terms) is a proof of necessity for infinite balance between both classes. This paper also explores Sophie Germain primes of the form where, is a prime. By extension, the paper also explores other properties of prime numbers. I derive a trigonometric product identity that isolates the condition for a prime p to form a Sophie Germain prime pair, i.e. such that is also prime. The analysis shows that the classical tangent–sine product expansion, when modified by the divisor-sum function, sigma(n), reproduces a constant equality only under this primality constraint.