E51: “Can least preimages of infinitely many totients have unbounded ratio?”

Open

Statement	User	Model	Harness	Time
2)V2Every value attained by Euler's totient function has a least natural-number preimage.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1There is an infinite set A of attained Euler-totient values such that the least n with φ(n)=a satisfies n/a →…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every value attained by Euler's totient function has a least natural-number preimage.

kernel-checked, filed Tue Aug 25 2026 04:06:38 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All natural a in the image of Euler's totient; existence and global minimality of a preimage.

1) V1 There is an infinite set A of attained Euler-totient values such that the least n with φ(n)=a satisfies n/a → ∞ as a tends to infinity through A.

open, filed Tue Aug 25 2026 04:06:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Root canonical statement. IsLeast encodes both attainment and global minimality. Tendsto from the subtype A at atTop faithfully formalizes a→∞ through the selected infinite values; the real cast makes the quotient ordinary real division.

Scope. One infinite A ⊆ ℕ; for each a in A, n(a) is the least natural-number preimage under Euler's totient; the real ratio n(a)/a tends to +∞ along the induced order on A.

E51:Erdős Problem #51 “Can least preimages of infinitely many totients have unbounded ratio?”

E51: “Can least preimages of infinitely many totients have unbounded ratio?”