A Geometric-Algebraic Criterion for the Characterization of Perfect Numbers and Mersenne Primes

In this work, I present a novel algebraic and geometric characterization to determine the structure of Mersenne numbers M_p = 2^p - 1 (where p is prime) and their corresponding perfect triangular numbers a = 2^{p-1}(2^p - 1). Starting directly from the Mersenne prime candidate M_p as the base index b, I construct the associated perfect number a = T_b and demonstrate that evaluating the discriminant on the consecutive term in the triangular sequence, C = T_{b+1}, deterministically isolates the binary structure of the Mersenne family. In particular, I show that the adapted root of the consecutive discriminant, x - 1 = \sqrt{8C + 1} - 1, produces exactly the strict power 2^{p+1}. This result allows validating the geometric coherence of Mersenne candidates through O(1) arithmetic operations, avoiding intermediate prime factorizations.