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Every even integer greater than 2 is a prime plus a product of at most 18 primes

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Wren Aquila ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

We prove that every even integer N≥4N \ge 4 can be written as N=p+ηN = p + \eta with pp prime and η≥2\eta \ge 2 a product of at most 1818 primes, counted with multiplicity. The best previously stated unconditional value we have located is due to Johnston and Starichkova: 395395 in the latest version of their paper, which revised the 369369 of its first two arXiv versions. Assuming that no Dirichlet LL-function vanishes in the half-plane Re⁡s>7/8\operatorname{Re} s > 7/8, the same holds with at most 1414 primes, and under the generalised Riemann hypothesis for Dirichlet LL-functions with at most 99. The improvement comes from counting rather than bounding: an explicit lower bound for the number of primes pp with N−pN - p free of small primes is played against explicit Hardy--Ramanujan-type upper bounds for the integers with many prime factors, both in the range below the sieve threshold and inside the sieved sets. These upper bounds weight by Chebyshev's identity log⁡n=∑d∣nΛ(d)\log n = \sum_{d \mid n} \Lambda(d), use the exact power-series coefficients of a majorant of a finite Euler product and the Mertens density of rough primes, and pre-sieve modulo 11551155, so that the integers to be excluded inherit the residue classes of the primes that produce them. All numerical steps are certified in interval or ball arithmetic, except some discrete convolutions with nonnegative terms, which are evaluated in binary64 under an explicit rounding-error bound and were also recomputed independently in ball arithmetic.

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Version history

  1. v1 · 19 pagesSubmitted by Wren AquilaInitial depositCurrentNov 13, 2026