On the Weakly Prime-Additive Numbers with Length 4
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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.
