Problem 1005

Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots$ be the Farey fractions of order $n\geq 4$ . Let $f(n)$ be the largest integer such that if $1\leq k<l\leq k+f(n)$ then $\frac{a_k}{b_k}$ and $\frac{a_l}{b_l}$ are similarly ordered - in other words,

(a_k-a_l)(b_k-b_l)\geq 0.

Estimate $f(n)$ - in particular, is there a constant $c>0$ such that $f(n)=(c+o(1))n$ for all large $n$ ?

Source: erdosproblems.com/1005

Number theory

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SolvedAnswered

An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Solved, in the site's label, which marks a resolution other than a proof or disproof, here an estimate: the answer to the question is yes, with $c=1/4$ . The upper half, $f(n)\le\lfloor n/4\rfloor+d$ with $d\in\{1,2,2,4\}$ by $n\bmod4$ , is van Doorn's Theorem 1 (arXiv:2509.00121v1, 28 August 2025; a preprint with no journal record found), an accepted partial claim on its page; the lower half, $f(n)\ge(\frac14-o(1))n$ , is Theorem 1 of the Cipollini preprint (arXiv:2607.23302v1, 25 July 2026), the accepted full claim, on its page, whose author declares that the manuscript was written by GPT-5.5 Pro, that its findings and strategy are due to the author together with GPT-5.5 Thinking, and that Aristotle, an automated proof system, and van Doorn produced a Lean 4 formalization. The site accepted the preprint as the resolution (its page was last edited 1 September 2026 and its commentary states $f(n)=(\frac14+o(1))n$ ). This is a source-supported solution accepted by the site, distinct from a claim of journal refereeing. Three external Lean developments proving the theorem for the papers' convention are recorded at pinned commits; they were not built or independently audited here, and no local kernel credit is claimed. Before Cipollini's preprint the truth was known to lie between $(\frac1{12}-o(1))n$ and $\frac n4+O(1)$ ; Erdős's 1943 bound was $f(n)\ge n/400-1$ , with $c=400$ read from his printed thresholds for $k\ge3$ . The frontmatter standing is derived from the claim pages; the exact-formula claim of Wang, Xie and Zhao, filed on the site's tab on 28 July 2026, stays a pending full claim on its page.

Er43: Erdős, P., A note on Farey series. Quart. J. Math. Oxford Ser. 14 (1943), 82--85, doi:10.1093/qmath/os-14.1.82 (Crossref record; received 30 March 1943). The Theorem, p. 82; the thresholds $n>192k$ and $n>400k$ , pp. 83--84 (the Rényi archive scan). Library home: Erdős 1943.
Ma42: Mayer, A. E., A mean value theorem concerning Farey series. Quart. J. Math. Oxford Ser. 13 (1942), 48--57, doi:10.1093/qmath/os-13.1.48 (Crossref record). Not held (publisher paywall); not requested here. The site credits it with $f(n)\to\infty$ ; van Doorn (p. 1) credits it with $f(n)\ge3$ for $n\ge5$ and attributes $f(n)\to\infty$ to Mayer's second 1942 paper, "On neighbours of higher degree in Farey series", Quart. J. Math. 13 (1942), 185--192, which is also the paper Erdős's footnote cites; neither is held, and the discrepancy is recorded, not resolved.
vD25b: W. van Doorn, Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions. arXiv:2509.00121v1 (28 August 2025), 9 pp. Theorem 1 and the Conjecture, p. 2; Theorem 2, p. 5. Library home: Doorn 2025.
Ci26: Cipollini, R., Optimality of Wouter van Doorn's Upper Bound for the Mayer--Erdős Farey Problem. arXiv:2607.23302v1 (25 July 2026), 16 pp. Theorem 1, p. 2; the contribution statement, p. 16. The key is identified with this paper by the page's own sentence and by the proof-claims tab's link to its arXiv identifier. Library home: Cipollini 2026.
WXZ26: Wang, Y., Xie, M. and Zhao, Z., An exact formula for Erdős' problem 1005. arXiv:2608.15681v1 (16 August 2026), 9 pp. Theorems 1.2 and 1.3, p. 1. A lead, not cited by the site's page; the first author's proof claim is on the site's tab. Library home: Wang and Zhao 2026.
Za06: Zaharescu, A., The Mayer--Erdős phenomenon. Indag. Math. (N.S.) 17 (2006), 147--156; [MeZa14] Meng, X. and Zaharescu, A., A multivariable Mayer--Erdős phenomenon. J. Korean Math. Soc. 51 (2014), 1029--1044. Not held; cited as van Doorn cites them (generalizations to linear forms, with the constant $1/480$ for [Za06]).
OEIS: van Doorn, W., Sequence A386893, "Minimal number of Farey fractions in between two fractions that are not similarly ordered", The On-Line Encyclopedia of Integer Sequences (created 9 September 2025; entry last modified 12 September 2025, server time; accessed through its JSON record), with the formula lines $a(n)<n/4+5$ and $a(n)\ge(n/12)(1-4/n^{1/3})$ and van Doorn's conjecture.

Problem 1005

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