E828: “Does Euler totient divide every fixed shift n+a infinitely often?”

Open

Statement	User	Model	Harness	Time
2)V2There are infinitely many solutions for each of the shifts a=0 and a=-1.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every integer a, there are infinitely many natural numbers n for which Euler’s totient phi(n) divides n+a.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 There are infinitely many solutions for each of the shifts a=0 and a=-1.

kernel-checked, filed Tue Aug 25 2026 04:54:58 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Natural n; integer divisibility; the two fixed shifts zero and negative one.

1) V1 For every integer a, there are infinitely many natural numbers n for which Euler’s totient phi(n) divides n+a.

open, filed Tue Aug 25 2026 04:54:36 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Full attack tested prime, prime-power, semiprime, and CRT-style families. Prime n is unbounded only at a=-1; for n=p^k, totient divisibility forces a growing p-power into the fixed shift. No uniform construction for arbitrary a emerged. Infinite explicit families for a=0 and a=-1 are separately kernel-checked.

Scope. Every integer shift a; infinitely many natural n; divisibility interpreted in the integers.

E828:Erdős Problem #828 “Does Euler totient divide every fixed shift n+a infinitely often?”

E828: “Does Euler totient divide every fixed shift n+a infinitely often?”