An Incremental Euler–Fermat Prime-Generation Machine: Modular Scheduling, Residual Prime Extraction, and Finite Validation

Abstract

This work presents an incremental prime-generation machine based on Euler’s polynomial (E(n)=n^2+n+41), the base-2 Fermat test, and a modular schedule of previously discovered factors. The method distinguishes between direct primes produced by the polynomial and residual primes extracted from composite values, without using conventional primality tests inside the generation process. It is proved that, for (n>40), after removing all previously known prime factors from (E(n)), the remaining cofactor is necessarily either 1 or a single prime number. Direct acceptance of values satisfying the Fermat congruence is formulated under an Euler–Fermat conjecture supported by finite computational evidence. In a validation up to (n=1{,}000{,}000), the machine generated 786,291 distinct primes, including 261,081 direct Euler primes and 525,210 residual primes extracted from composite values. External validation found no pseudoprimes, composite residuals, or classification errors. The study clearly separates proved results, conjectural statements, and computational evidence.