Butler 2015 egyptian fractions each denominator having three

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theorem_1: Every natural number is a sum of distinct unit fractions whose denominators are each a product of three distinct primes.

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Butler, Steve and Erdős, Paul and Graham, Ron, Egyptian fractions with each denominator having three distinct prime divisors. Integers 15 (2015), Paper No. A51, 9 pp. The file prints no license; the journal's site states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License so that all content is freely available without charge to the users or their institutions." (https://math.colgate.edu/~integers/, read 2026-10-02): the Creative Commons Attribution 4.0 license.

Theorem 1 states that every natural number is an Egyptian fraction (a sum of reciprocals of distinct integers) all of whose denominators are products of three distinct primes; Theorem 3 strengthens this to three distinct odd primes. The result had been attributed to Erdos and Graham in Guy's problem book (D11) in a stronger square-free form but never published, and this note supplies the natural-number case. The method is to show that sums of products of primes cover a long contiguous interval of integers: Lemma 1 shows that for n >= 5 the subset-sum set L_{n+3}(n) contains every integer from (1/6) to (5/6) of the full sum sigma_{n+3}(n), and the argument invokes Olson's theorem that the subset sums of a set X of distinct nonzero residues mod a prime p with p < (|X|^2+3)/4 cover every residue, applied to products of all but one of the first n+2 primes. The paper also conjectures the analogous statement for two distinct prime divisors, noting Johnson's expression of 1 as a sum of 48 unit fractions with two-prime denominators, and remarks that similar arguments give denominators with any fixed number l >= 4 of distinct primes. This is the reference for problem 306 on Egyptian fractions whose denominators have a prescribed number of prime divisors.

Source: https://math.colgate.edu/~integers/vol15.html.

The retained folder-name PDF is the journal's typeset article, Integers 15 (2015), paper A51, 9 pages (received 2/21/15, accepted 11/17/15, published 12/11/15). Read status: claims checked. Theorem 1, Theorem 3, Lemma 1, Olson's theorem as quoted and the remarks of pp. 2 and 8 were read clause by clause (pp. 2 and 8 on the page images, the rest in the text layer); the proof of Theorem 1 was read for structure and the proof of Lemma 1 was not read. Result page: theorem_1.

Bears on. #306

Results to transcribe.

• Theorem 1 (p. 2): "Any natural number can be written as an Egyptian fraction where each denominator is the product of three distinct primes."
• Theorem 3 (p. 8): "Any natural number can be written as an Egyptian fraction where each denominator is the product of three distinct odd primes."
• Lemma 1: For n >= 5, the subset-sum set L_{n+3}(n) contains every integer i with (1/6)sigma_{n+3}(n) <= i <= (5/6)sigma_{n+3}(n), giving a long contiguous interval of sums of products of primes.
• Olson's theorem (quoted as Theorem 2): If X is a set of distinct nonzero residues mod a prime p and p < (|X|^2+3)/4, then the set P(X) of subset sums of X contains all residues mod p.
• Conjecture / remark: The same should hold with denominators having two distinct prime divisors; similar arguments handle l >= 4 distinct primes.