Doorn 2025 smallest denominator not contained unit fraction

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inequality_1_2: Bounds the least missing denominator by the number of k-term representations of one, giving v(k) at most the Vardi constant to the power (2/5 + o(1)) 2^k (the paper prints 1/5).

section_3: Records the paper's concluding remarks tying the least missing denominator v(k) to the least number N(b−1,b) of unit fractions representing (b−1)/b, in both directions.

theorem_1_1: The least integer above one missing from every k-term representation of one by distinct unit fractions is at least exp(c k²) for an absolute c.

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Wouter van Doorn, Quanyu Tang, The smallest denominator not contained in a unit fraction decomposition of $1$ with fixed length. arXiv:2512.22083 (2025). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2512.22083), every other right reserved.

The copy read for this card is arXiv:2512.22083v2 (24 May 2026; v1 is dated 26 December 2025), seven pages with a clean text layer, whose comments line says the paper was accepted for publication in Mathematical Proceedings of the Cambridge Philosophical Society with minor revisions following the referee's suggestion. The version of record appeared online on 8 July 2026 (Math. Proc. Cambridge Philos. Soc., pp. 1--9, doi:10.1017/S0305004126102102). The published text was not compared with v2; every locator on this card and its result pages refers to v2.

Let D_k be the set of denominators occurring in some decomposition 1 = 1/n_1 + ... + 1/n_k with distinct 1 <= n_1 < ... < n_k, and let v(k) be the least integer above 1 not in D_k; Erdos and Graham asked for the growth of v(k), suggested v(k) >> k! and speculated on doubly exponential growth. Van Doorn and Tang prove Theorem 1.1: there is an absolute c > 0 with v(k) >= e^{c k^2} for all k, the first lower bound in the literature (the authors add (p. 2) that even v(k) >> k! does not look easy to them to derive from the Bleicher-Erdos papers). The argument first proves Lemma 2.1, the nesting D_k contained in D_{k+1}, by case analysis on the largest denominator using the splitting identity 1/n = 1/(n+1) + 1/(n(n+1)) and its variant 1/ab = 1/(ab+a) + 1/(b(ab+a)) for composite denominators, and then applies Vose's bound N(b) << (log b)^{1/2} for the least number of unit fractions needed to write a/b. For the upper side, v(k) <= k F(k) + 2 together with Elsholtz and Planitzer's count of decompositions gives v(k) <= c_0^{(1/5 + o(1)) 2^k} with c_0 the Vardi constant. The paper's c_0 = 1.264085... (footnote 1: OEIS A076393) with exponent (1/5 + o(1)) 2^k is the site's form of the upper bound for problem 148; Elsholtz and Planitzer's Corollary 3(2) prints f_k(1,1) < c_0^{(2/5 + epsilon) 2^{k-1}} with their c_0 = lim u_n^{2^{-n}} = 1.5979..., u_0 = 1, u_{n+1} = u_n(u_n + 1), the square of the Vardi constant, which is the Vardi constant to the power (2/5 + epsilon) 2^k, twice the exponent written here; the difference is recorded on the problem 148 page and on the elsholtz_2021 corollary_3 page. The paper thereby advances problem 293, and Section 3 explains its link with problem 304 on N(b): if the conjecture N(b) << log log b holds, the authors expect (p. 6) that the lower bound could then be raised to e^{e^{ck}}, matching the doubly exponential guess (an expectation, not a theorem), while conversely b < v(k) implies N(b-1, b) <= k - 1, so lower bounds for v give upper bounds for N.

Source: https://arxiv.org/abs/2512.22083.

Read status: claims checked. Theorem 1.1, Lemma 2.1, Lemma 2.2 (Vose's theorem as restated), inequality (1.2) with the derived upper bound and the statements of Section 3 were read clause by clause in the text layer of v2; the proof of Theorem 1.1 (pp. 4--6) was read for structure and summarized, not verified. Result pages: theorem_1_1, inequality_1_2, section_3.

Bears on. #293, #304, #148 (inequality (1.2), v(k) <= k F(k) + 2, and the form of the upper bound on F(k))

Results to transcribe.

• Theorem 1.1: There is an absolute c > 0 with v(k) >= e^{c k^2} for all positive integers k, the first known lower bound on the smallest denominator missing from all k-term decompositions of 1.
• Lemma 2.1: D_k is contained in D_{k+1} for all k >= 2, so the sets of usable denominators are nested.
• Upper bound: v(k) <= k F(k) + 2 combined with Elsholtz and Planitzer's bound gives v(k) <= c_0^{(2/5 + o(1)) 2^k}, with c_0 the Vardi constant (the paper prints 1/5; see the paragraph above).
• Link to N(b): Using Vose's N(b) << (log b)^{1/2} drives the main proof; the conjectural N(b) << log log b would, the authors expect, give v(k) >= e^{e^{ck}}, and conversely b < v(k) implies N(b-1,b) <= k - 1.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.