Problem 417

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claims/: The 0 claim pages of Problem 417, one per claimant's result; the problem's standing derives from them.

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Statement. Let

$$V'(x)=\#\{\phi(m) : 1\leq m\leq x\}$$

and

$$V(x)=\#\{\phi(m) \leq x : 1\leq m\}.$$

Does $\lim V(x)/V'(x)$ exist? Is it $>1$?

Formulation. The second question, whether the limit is $>1$, presupposes the first. It is read as the formal-conjectures statement linked under Formalization states it (erdos_417.parts.ii): whether $V(x)/V'(x)$ tends to a limit $L>1$, possibly infinite, so a yes to it includes a yes to the first.

Status. Open. The site labels the problem OPEN, and its proof-claims thread carried no claim as of 2026-10-06; no claim page is recorded: no one claims either question. Theorem 2.2 of the OpenAI release manuscript of 25 September 2026 bears on the problem without claiming it; the Current assessment records the item and why it is not a claim. The derived standing is open.

Source. erdosproblems.com/417, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #417, https://www.erdosproblems.com/417.

References.

• [Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.

Formalization. Statement in the file ErdosProblems/417.lean of formal-conjectures, pinned at its last change of 18 September 2026: erdos_417.parts.i (the existence of the limit) and erdos_417.parts.ii (a limit exceeding $1$), each an answer(sorry) equivalence marked research open with no formal_proof attribute.

Current assessment

The question (site formulation, 2026-09-04). Whether $\lim V(x)/V'(x)$ exists and whether it exceeds $1$. OPEN. The site's commentary records that $V'(x)\le V(x)$ trivially and that Erdős [Er98] suggested the limit may be infinite.

Standing. No claim page: no one claims either question, and the derived standing is open.

The OpenAI release item (family 024). Theorem 2.2 of the release manuscript of 25 September 2026, An asymptotic formula for the number of totients (intake card, Theorem 2.2 page), claims, for each fixed $k\ge1$, an asymptotic for $N_k(x)=\#\{v\le x\text{ totient}:kx<\ell(v)\le(k+1)x\}$, with $\ell(v)$ the least preimage of $v$, and that $N_k(x)$ is of order $V(x)$ for $k=1$ and $2$; the manuscript is unrefereed, and this corpus has not built the release's Lean for these counts (weighted_totient_asymptotic, weighted_totient_one_two). The manuscript does not mention $V'$ or this problem, and the release claims nothing about it, so the item is recorded here and not as a claim.

Search scope (2026-10-06 and 2026-10-07). The site's page and its proof-claims thread (empty), the formal-conjectures statement file and the OpenAI release at the pinned revision of its manuscript; no other literature search was made, and no proof was independently assessed.

Known Results

The ratio $V(x)/V'(x)$ is bounded: Pollack, Pomerance and Treviño quote $V(x)\asymp V'(x)$ from Ford on p. 7 of their manuscript, which excludes the infinite limit the site reports Erdős suggested.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• openai_2026_asymptotic_formula_number_totients
• openai_2026_asymptotic_formula_number_totients / theorem_2_2