What question did this study set out to answer?

To establish a Wilson–Fermat resonance equation and explore its implications for primes and composite numbers.

March 10, 2026Open Access

A Wilson–Fermat Resonance Equation for Primes

Key Points

To establish a Wilson–Fermat resonance equation and explore its implications for primes and composite numbers.
Introduced the product X(n) = ((n − 1)! + 1)(2^(n−1) − 1) for n ≥ 3.
Decomposed X(n) relative to the lattice n²Z as X(n) = Λ(n) n² + ε(n).
Defined the error function A(n) = |ε(n)| and analyzed its behavior for primes and composite values.
For primes p, A(p) = 0 indicating exact resonance.
Identified small values of A(n) as reflective of a resonance landscape for composites.
Observed that powers of two show a quasi-resonance with A(4) = 1 and A(2^m) = 1 for m ≥ 4.

Abstract

We introduce a Wilson–Fermat resonance equation built from the productX (n) = ( (n − 1) ! + 1) (2^ (n−1) − 1), n ≥ 3. Instead of studying only divisibility, we decompose X (n) relative to the lattice n2Z asX (n) = Λ (n) n² + ε (n), where Λ (n) ∈ Z and ε (n) is the centered minimal additive correction. This yields an errorfunctionA (n) = |ε (n) |. For primes p, one has A (p) = 0. Computational evidence suggests that small values of A (n) encode a nontrivial resonance landscape for composite numbers. In particular, powers of twoexhibit an exact quasi-resonance: A (4) = 1, A (2ᵐ) = 1 for all m ≥ 4. This motivates the viewpoint that primality corresponds to exact Wilson–Fermat resonance, while certain composite families occupy the first nonzero resonance level.

A Wilson–Fermat Resonance Equation for Primes

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Abstract

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