Cubic Relations of Mersenne and Fermat Numbers as a Consequence of Ramanujan's Polynomial Identities

This paper introduces a novel approach to Diophantine equations by applying an exponential substitution (k=2^m) to Ramanujan's telescoping identities for sums of cubes. This method reveals a strict analytical identity (Vorobtsov's Corollary) linking the cubes of Mersenne numbers of different orders. The research demonstrates that these prime numbers are born not in a mathematical vacuum, but inside rigid polynomial cubic structures.

Furthermore, we show that the resulting polynomial matrix acts as a "universal prime incubator"— a combined algebraic sieve that simultaneously generates prime divisors from both the Mersenne and Fermat families within single composite blocks. This discovery not only enriches the reference apparatus for Mersenne and Fermat numbers but also serves as a striking example of interdisciplinary synergy, bridging the gap between abstract algebraic geometry and applied number theory, and opening new prospects for parallel prime search algorithms.