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February 11, 20260 citationsOpen Access

The Critical Resonance Band Theory for Mersenne Prime Exponents

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JMJose Manuel Briceno Mendoza

Key Points

  • To explore the shared characteristics of Mersenne prime exponents and to identify new candidates through a resonance band.
  • Analytical function developed from fundamental constants
  • Examination of all known Mersenne prime exponents
  • Establishment of a numerical band for resonances
  • All known Mersenne prime exponents fall within a narrow band (0.99 to just below 1)
  • Identification of a hidden pattern in Mersenne primes
  • The theory proposes a predictive filter for future candidate exponents

Abstract

We have discovered that all known Mersenne prime exponents share a precise "resonance" signature, measured via a function built from fundamental constants. Every known exponent falls within the same narrow numerical band (0.99 to just below 1), revealing a hidden pattern behind these special primes. The theory acts as a predictive filter, identifying new candidate exponents and offering a fresh mathematical perspective on Mersenne prime distribution.

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Cite This Study

Jose Manuel Briceno Mendoza (2026) studied this question.

synapsesocial.com/papers/698c1c22267fb587c655e4dehttps://doi.org/10.5281/zenodo.18529541
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