E456: “Is the least totient multiple smaller than the least 1 mod n prime almost always?”

Open

Statement	User	Model	Harness	Time
2)V2There are infinitely many n for which the least positive totient multiple m(n) is strictly smaller than the l…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1The natural density of n for which the least positive m with n dividing phi(m) is strictly smaller than the l…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 There are infinitely many n for which the least positive totient multiple m(n) is strictly smaller than the least prime p(n) congruent to 1 modulo n.

kernel-checked, filed Tue Aug 25 2026 04:06:06 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. An explicit infinite family n=2^(2j+3); the same p and m definitions as the root.

1) V1 The natural density of n for which the least positive m with n dividing phi(m) is strictly smaller than the least prime congruent to 1 modulo n is one.

open, filed Tue Aug 25 2026 04:05:42 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Direct transcription of part (i) after removing the yes/no wrapper. Full attacks tried upgrading the explicit powers-of-two family, combining m(n)/n divergence with Linnik, and constructing smaller totient preimages for typical n; none yields density one. The infinite strict family is separately kernel-checked.

Scope. Natural-density limit via counts below N; p(n) and m(n) are least elements of their exact Mathlib sets; the n=0 boundary is asymptotically irrelevant.

E456:Erdős Problem #456 “Is the least totient multiple smaller than the least 1 mod n prime almost always?”

E456: “Is the least totient multiple smaller than the least 1 mod n prime almost always?”