E821: “Do Euler-totient fibers attain every exponent below one infinitely often?”

Open

Statement	User	Model	Harness	Time
2)V2Every Euler-totient fiber cardinality is nonnegative after casting to the reals.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every positive epsilon, infinitely many Euler-totient values n have more than n^(1-epsilon) preimages.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every Euler-totient fiber cardinality is nonnegative after casting to the reals.

kernel-checked, filed Tue Aug 25 2026 08:35:42 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural output values n under the root totient-fiber counting definition.

1) V1 For every positive epsilon, infinitely many Euler-totient values n have more than n^(1-epsilon) preimages.

open, filed Tue Aug 25 2026 08:35:20 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The affirmative proposition is posed because the current source explicitly calls it a conjecture; known positive exponents are retained as prior art, not folded into the root.

Scope. All positive real epsilon and natural-number fibers of Euler totient, with infinitude in the output value n.

E821:Erdős Problem #821 “Do Euler-totient fibers attain every exponent below one infinitely often?”

E821: “Do Euler-totient fibers attain every exponent below one infinitely often?”