# On the Weakly Prime-Additive Numbers with Length 4

## TL;DR

This paper proves the existence of infinitely many length 4 weakly prime-additive numbers divisible by any positive integer m under GRH, extending previous results from length 3 to length 4.

## Contribution

It establishes the existence of infinitely many length 4 weakly prime-additive numbers divisible by any positive integer m under GRH, generalizing earlier length 3 results.

## Key findings

- Existence of infinitely many length 4 weakly prime-additive numbers divisible by any m under GRH.
- Extension of length 3 results to length 4 in weakly prime-additive numbers.
- Related results analogous to the length 3 case are also presented.

## Abstract

In 1992, Erd$\H{o}$s and Hegyv$\'{a}$ri showed that for any prime p, there exist infinitely many length 3 weakly prime-additive numbers divisible by p. In 2018, Fang and Chen showed that for any positive integer m, there exists infinitely many length 3 weakly prime-additive numbers divisible by m if and only if 8 does not divide m. Under the assumption (*) of existence of a prime in certain arithmetic progression with prescribed primitive root, which is true under the Generalized Riemann Hypothesis (GRH), we show for any positive integer m, there exists infinitely many length 4 weakly prime-additive numbers divisible by m. We also present another related result analogous to the length 3 case shown by Fang and Chen.