Alekseyev 2019 partitions into squares distinct integers whose

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lemma_2: For a finite set X of positive integers with s the sum of 1/x and n the sum of x^d, bounds |X| by s times the (d+1)th root of n/s and confines min X between the ceiling of 1/s and a floor of two roots, which bounds Alekseyev's exhaustive search.

theorem_1: States that every integer above 8542 is a sum of squares of distinct positive integers whose reciprocals sum to 1, and that 8542 is not.

theorem_6: Alekseyev's translation criterion: if S is a complete set of t-translations with maximum scale q and maximum shift s, and n+1, ..., qn+s are all t-representable, then every number greater than n is t-representable.

theorem_7: States that 15707 is the largest integer that is not a sum of squares of distinct integers, each at least 6, whose reciprocals sum to 1; every larger integer is such a sum.

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Alekseyev, Max A., On partitions into squares of distinct integers whose reciprocals sum to 1. (2019), 213--221.

The copy read for this card is arXiv:1801.05928v2 (23 April 2018; v1 18 January 2018), 7 pages, the latest version on the arXiv listing read. The citation's "(2019), 213--221" is the chapter in The Mathematics of Various Entertaining Subjects, Volume 3: The Magic of Mathematics (J. Beineke and J. Rosenhouse, eds.), Princeton University Press, 2019, pp. 213--221, DOI 10.2307/j.ctvd58spj.18, per the arXiv journal reference and the Crossref records read the same day; the published chapter was not obtained or compared, and the locators below are the preprint's. Read status: claims checked. The definition of a representable integer and Theorem 1 (p. 1) and Lemma 2 (p. 2) were read clause by clause in the text layer on 2026-09-18, and on 2026-10-08 again on the page images together with Lemmas 3--5 (pp. 3--5), the definitions of Section 3 and Theorems 6 and 7 (p. 6); the proofs were read for structure and the computations were not rerun. Result pages: Theorem 1 (p. 1), Lemma 2 (p. 2), Theorem 6 (p. 6) and Theorem 7 (p. 6). The arXiv record names arXiv's non-exclusive distribution license (arXiv:1801.05928), every other right reserved.

Call m representable if there are distinct positive integers x_1,...,x_k with 1/x_1+...+1/x_k = 1 and m = x_1^2+...+x_k^2. Theorem 1 states that the largest non-representable integer is 8542, which proves Graham's 1963 conjecture that all sufficiently large integers are representable and pins down the exact threshold. The proof generalizes Graham's method of translating representations of smaller numbers into representations of larger ones, introducing a class of translations acting on restricted representations that yields a second proof (Theorems 6 and 7: a complete set of $t$-translations reduces $t$-representability of all large integers to a finite range, and 15707 is the largest integer with no representation by integers at least 6); Lemma 2 gives power-mean bounds on |X| and on min X, which drive an exhaustive search algorithm used both to build small representations and to certify that 8542 is not representable. Problem 283 asks, for an integer polynomial p with positive leading coefficient whose values have no common divisor above 1, whether every large m is the sum of p over the denominators of some representation of 1 by distinct unit fractions; Theorem 1 settles the case p(x) = x^2 with the exact threshold, and the paper cites Graham's 1963 theorem for p(x) = x (every integer above 77 is a sum of distinct integers whose reciprocals sum to 1). For Problem 351, on the strong completeness of {p(n) + 1/n}, Theorem 1 gives the case p(x) = x^2 only without the removal of a finite set (see the result page).

Source: https://arxiv.org/abs/1801.05928.

Bears on.

• #283: Theorem 1 is the case $p(x)=x^2$ for every $m>8542$, and 8542 has no representation (Lemma 3); Theorem 7 gives that case for every $m>15707$ with all denominators at least 6; Lemma 2 and Theorem 6 bear only through these.
• #351: the paper does not treat this problem; with $m+1=\sum(x_i^2+1/x_i)$, Theorem 1 makes every integer $\ge8544$ a finite sum of distinct terms of $\{n^2+1/n\}$, the case $p(x)=x^2$ without the removal of a finite set that strong completeness requires.

Results to transcribe.

• Theorem 1: The largest integer not expressible as a sum of squares of distinct positive integers whose reciprocals sum to 1 is 8542; every larger integer is so expressible.
• Lemma 2: For a positive integer d and a finite set X of positive integers with s = sum 1/x and n = sum x^d, the power mean inequality gives |X| <= s (n/s)^{1/(d+1)} and ceil(1/s) <= min X <= floor(min{(n/s)^{1/(d+1)}, n^{1/d}}), bounding the exhaustive search.
• Theorem 6: For a complete set of t-translations with maximum scale q and maximum shift s, t-representability of n+1, ..., qn+s gives it for every number greater than n.
• Theorem 7: The largest integer that is not 6-representable is 15707.

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