What question did this study set out to answer?

The aim is to explore how the structure of integers in the Standard Model influences the number of quark-lepton generations.

April 5, 2026Open Access

Arithmetic Admissibility and the Number of Quark-Lepton Generations

Key Points

The aim is to explore how the structure of integers in the Standard Model influences the number of quark-lepton generations.
Analyzed structural integers of the Standard Model for prime factors up to 3.
Utilized Størmer's theorem regarding 3-smooth numbers.
Examined implications of the Kobayashi-Maskawa mechanism for CP violation.
Found that maximal quark-lepton generations are 4 under certain conditions.
Determined exactly 3 generations if neutrinos are Majorana.
Identified that the prime factor 5 limits both a fourth generation and SU(5) grand unification.

Abstract

I show that if the structural integers of the Standard Model have no prime factor exceeding 3 (arithmetic admissibility), then the number of quark-lepton generations is at most 4 unconditionally, and exactly 3 if neutrinos are Majorana. The proof uses Størmer's theorem (1897) on consecutive 3-smooth numbers and the Kobayashi-Maskawa mechanism for CP violation (1973). The same arithmetic obstruction (the prime 5) that excludes a fourth generation also excludes SU(5) grand unification and the minimal supersymmetric standard model.

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