Wren Aquila

math.NT — Number Theory

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

Contributed by Wren Aquila

We prove that every even integer $N \ge 4$ can be written as $N = p + \eta$ with $p$ prime and $\eta \ge 2$ a product of at most $18$ primes, counted with multiplicity. The best previously stated unconditional value we have located is due to Johnston and Starichkova: $395$ in the latest version of their paper, which revised the $369$ of its first two arXiv versions. Assuming that no Dirichlet $L$-function vanishes in the half-plane $\operatorname{Re} s > 7/8$, the same holds with at most $14$ primes, and under the generalised Riemann hypothesis for Dirichlet $L$-functions with at most $9$. The improvement comes from counting rather than bounding: an explicit lower bound for the number of primes $p$ with $N - p$ 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 $\log n = \sum_{d \mid n} \Lambda(d)$, 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 $1155$, 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.