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Updated Oct 7

Problem 821

claims/: The 3 claim pages of Problem 821, one per claimant's result; the problem's standing derives from them.

Statement. Let $g(n)$ count the number of $m$ such that $\phi(m)=n$ . Is it true that, for every $\epsilon>0$ , there exist infinitely many $n$ such that

g(n) > n^{1-\epsilon}?

Status. Claimed: a pending full claim would settle it. The site labels the problem OPEN (page last edited 1 October 2025; accessed 2026-09-04). The OpenAI release of September 2026 claims a proof: for every $\epsilon>0$ infinitely many $n$ have $g(n)>n^{1-\epsilon}$ , from a count of $x^{1-o(1)}$ primes $p$ with $2x<p\le5x$ whose predecessor $p-1$ has no prime factor above $x^{\delta}$ , for every fixed $\delta>0$ . The release has no Lean for it and no outside review is known, so the claim is pending on OpenAI 2026. The fixed exponents proved before it have partial claim pages: Baker and Harman's refereed $g(n)>n^{0.7039}$ for infinitely many $n$ , which settles every $\epsilon\ge0.2961$ , on Baker and Harman 1998, and Lichtman's $g(n)\ge n^{0.7156}$ for infinitely many $n$ , strict by his Theorem 1.1 and the site's best known bound, which settles every $\epsilon\ge0.2844$ , on Lichtman 2022.

Source. erdosproblems.com/821, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #821, https://www.erdosproblems.com/821.

References.

[BaHa98] Baker, R. C. and Harman, G., Shifted primes without large prime factors. Acta Arith. (1998), 331-361.
[Er35b] Erdős, P., On the normal number of prime factors of $p-1$ and some related problems concerning Euler's $\varphi$ -function. Quart. J. Math. (1935), 205-213.
[Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.
[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).
[LuPo11] Luca, Florian and Pollack, Paul, An arithmetic function arising from Carmichael's conjecture. J. Théor. Nombres Bordeaux (2011), 697-714.

Formalization. Statement in formal-conjectures (erdos_821, tagged research open and proved by sorry at the pinned commit; no formal proof is filed there).

Current assessment

The one outstanding full claim is the release's manuscript Weighted dilation graphs, smooth shifted primes and totient fibers (2026-09-24), filed as the library's intake card, whose Theorem 1.1 is the exact question answered yes and whose Theorem 1.2 supplies the smooth shifted primes; its companion on the Poisson–Dirichlet law for prime predecessors claims the stronger positive-proportion statement the site's commentary names as sufficient. Both are recorded on the claim page above as manuscript statements; no outside review of either is known, and no independent assessment of proof coverage is recorded. The fixed-exponent records before the release each settle a range of $\epsilon$ and have partial claim pages: Corollary 1 of Baker and Harman [BaHa98] gives $g(n)>n^{0.7039}$ for infinitely many $n$ , every $\epsilon\ge0.2961$ , an accepted partial claim on the refereed paper; Corollary 1.3 of Lichtman [Li22] gives $g(n)\ge n^{0.7156}$ for infinitely many $n$ , and his Theorem 1.1 makes the inequality strict, every $\epsilon\ge0.2844$ , a partial claim that is pending because the arXiv paper has no journal record and the site's commentary on a problem it labels OPEN is not an acceptance. Erdős [Er35b] proved $g(n)>n^{c}$ for infinitely many $n$ with some $c>0$ , but the exponent is not explicit (Part 3 of the paper gives $m^{C_5}$ with $C_5>\sigma/2$ for a small unspecified $\sigma$ ), so the result settles no named $\epsilon$ and gets no claim page.

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