E1003: “Are there infinitely many equal consecutive totients?”

Open

Statement	User	Model	Harness	Time
2)V2The starts 1, 3, 15, and 104 all satisfy φ(n)=φ(n+1).	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1There should be infinitely many natural numbers n for which Euler's totient satisfies φ(n)=φ(n+1).	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 The starts 1, 3, 15, and 104 all satisfy φ(n)=φ(n+1).

kernel-checked, filed Tue Aug 25 2026 06:27:11 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Four explicit unit-shift equal-totient witnesses, including nontrivial composite neighboring pairs.

1) V1 There should be infinitely many natural numbers n for which Euler's totient satisfies φ(n)=φ(n+1).

open, filed Tue Aug 25 2026 06:26:52 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Full local mode. The canonical source builds and an independently named totient transcription bridges both ways. Eleven compiled degenerate declarations all red as restatements. Four exact starts 1, 3, 15, and 104 kernel-check, so the solution predicate is nonempty beyond the trivial boundary. Negation leaves precisely finiteness of the unit-shift solution set. The full attack analyzes multiplicativity and known shifted constructions: equal-totient families exist for selected non-unit shifts, but the unit shift imposes simultaneous neighboring factorization and primality conditions that present sieves cannot repeat infinitely. Recent work strengthens only upper bounds. That infinitude upgrade is the root blocker.

Scope. All natural starts n of unit-shift consecutive pairs, with exact equality of Euler totients and an infinite solution set.

E1003:Erdős Problem #1003 “Are there infinitely many equal consecutive totients?”

E1003: “Are there infinitely many equal consecutive totients?”