E479: “Does every residue k>1 occur infinitely often as 2^n mod n?”

Open

Statement	User	Model	Harness	Time
2)V2There are infinitely many n such that 2^n≡4 modulo n.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every natural number k>1, infinitely many n satisfy 2^n≡k modulo n.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 There are infinitely many n such that 2^n≡4 modulo n.

kernel-checked, filed Tue Aug 25 2026 06:30:17 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The complete residue k=4 case of Erdős 479.

1) V1 For every natural number k>1, infinitely many n satisfy 2^n≡k modulo n.

open, filed Tue Aug 25 2026 06:29:03 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Full-local mode. Twelve compiling attacks are red for restatement; k=4,n=6 supplies a concrete instance; independent transcription is equivalent; direct negation and clean exact? fail. Searches confirm known infinitude only for special residues including powers of two, -1, -2, and 0. Whole routes examined Fermat congruences, CRT, pseudoprime constructions, primitive divisors, witness-lifting, and the k=3 computational barrier. Lean proves k=4 using the infinite family n=2p over odd primes: Fermat gives 2^(2p)≡4 mod p, parity gives the congruence mod 2, and CRT combines them. No Commons or computational exhaustion.

Scope. The positive-answer right side of current Formal Conjectures erdos_479; its binder `∀ᵉ (k > 1)` is transcribed as the ordinary bounded universal `∀ k > 1`.

E479:Erdős Problem #479 “Does every residue k>1 occur infinitely often as 2^n mod n?”

E479: “Does every residue k>1 occur infinitely often as 2^n mod n?”