Elsholtz 2020 number solutions erdos straus equation

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corollary_1: The Erdős–Straus equation 4/n = 1/a1 + 1/a2 + 1/a3 has at most O_ε(n^(3/5+ε)) solutions in positive integers for every n, extending the Elsholtz–Tao bound from prime to arbitrary denominators.

corollary_3: Bounds f_k(1,1), the number of nondecreasing k-tuples of positive integers whose reciprocals sum to 1, by k^((7/51) 2^(k-1) + ε) and, for large k, by c_0^((7/17 + ε) 2^(k-1)), and bounds the solutions of 1 = Σ 1/a_i + 1/Π a_i.

theorem_1: For any m and n the equation m/n = 1/a1 + 1/a2 + 1/a3 has at most O_ε(n^ε (n^3/m^2)^(1/5)) solutions in positive integers.

theorem_2: Bounds the number f_k(m,n) of representations of m/n as a sum of k unit fractions: f_4(m,n) << n^ε (n^(4/3)/m^(2/3) + n^(28/17)/m^(8/5)), and for k ≥ 5 f_k(m,n) << (kn)^ε (k^(4/3) n^2/m)^((28/17) 2^(k-5)).

theorem_3: For each m there are infinitely many n with f_3(m,n) at least exp((log 6 + o(1)) log n / log log n), a density-one set of n with f_3(m,n) at least (log n)^(log 3 + o(1)), and for m = 4 a density-one set with f_3(4,n) at least (log n)^(log 6 + o(1)).

theorem_4: For every m and every reduced residue class e mod f there are infinitely many primes p ≡ e mod f with f_3(m,p) >>_{f,m} exp((5 log 2/(12 lcm(m,f)) + o(1)) log p/log log p).

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Elsholtz, Christian and Planitzer, Stefan, The number of solutions of the Erdős-Straus equation and sums of $k$ unit fractions. Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 3, 1401--1427.

Theorem 1 shows that for all m, n and any epsilon > 0 the equation m/n = 1/a1 + 1/a2 + 1/a3 has at most O_eps(n^eps (n^3/m^2)^{1/5}) solutions in positive integers, improving Browning and Elsholtz's O_eps(n^eps (n/m)^{2/3}) in the range m much less than n^{1/4}. Corollary 1 specializes this to the Erdős-Straus equation 4/n = 1/x + 1/y + 1/z, giving at most O_eps(n^{3/5+eps}) solutions for arbitrary denominators n and so extending the Elsholtz-Tao bound, previously known only for n prime. Corollary 2 gives an algorithm listing all such representations in expected time O_eps(n^eps (n^3/m^2)^{1/5}), and Theorem 2 bounds f_4(m,n) by O_eps(n^eps(n^{4/3}/m^{2/3} + n^{28/17}/m^{8/5})), with a corresponding bound for k at least 5. The abstract also records improved lower bounds: for every m and every reduced residue class e mod f there are infinitely many primes p in the class e mod f with the number of solutions of m/p = 1/a1 + 1/a2 + 1/a3 of order >>{f,m} exp((5 log 2/(12 lcm(m,f)) + o{f,m}(1)) log p / log log p) (Theorem 4), where the previous best lower bound of this type was of order (log p)^{0.549}. The methods are parametrizations of the solution set combined with divisor-function estimates. For Problem 242 the paper gives counting bounds for solutions of the Erdős-Straus equation, upper (Corollary 1) and lower (Theorems 3 and 4); for Problem 148, Corollary 3 bounds the number of representations of 1 as a sum of k unit fractions from above.

Source: https://arxiv.org/abs/1805.02945.

The copy read for this card is arXiv:1805.02945v1 (8 May 2018, 21 pages; the only version listed on 2026-09-18); the published version, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 3, 1401--1427, DOI 10.1017/prm.2018.137, online 30 January 2019 (Crossref record fetched), was not compared. Read status: claims checked. Theorems 1--4 and Corollaries 1--4 were read clause by clause on the page images of pp. 1--4, and Remarks 4 and 5 on p. 20. The proofs of Theorems 1--4 (Sections 5--7, pp. 8--20) were read for their structure only and were not checked step by step. Theorem 3 (pp. 3--4) carries the lower bounds for general denominators n, among them f_3(4,n) >= exp((log 6+o(1)) log log n) for n in a set of density one, and Theorem 4 (p. 4) the lower bound for prime denominators that the abstract states. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1805.02945), every other right reserved.

Bears on.

• #242: Corollary 1 bounds the number of solutions of 4/n = 1/a1 + 1/a2 + 1/a3 by O_eps(n^{3/5+eps}) for every n; Theorem 3 gives, for n in a set of density one, f_3(4,n) >= (log n)^{log 6+o(1)}, and Theorem 4 gives, in each reduced residue class e mod f, infinitely many primes p with f_3(4,p) >>_f exp((5 log 2/(12 lcm(4,f)) + o_f(1)) log p/log log p). Upper bounds and lower bounds on a density-one set or on infinitely many primes do not decide whether every n > 2 has a solution.
• #148: Corollary 3 bounds f_k(1,1), the number of nondecreasing k-tuples with reciprocal sum 1, which is at least the problem's F(k); part 2 prints its constant c_0 from a sequence started at u_0 = 1, while p. 1 starts it at u_1 = 1. The paper proves no lower bound for F(k); p. 1 cites Konyagin's.

Results. Page numbers are those of arXiv v1 (pp. 1--21).

• Theorem 1 (p. 2; proof pp. 9--10): for all m, n and eps > 0, m/n = 1/a1 + 1/a2 + 1/a3 has at most O_eps(n^eps (n^3/m^2)^{1/5}) solutions in positive integers.
• Corollary 1 (p. 2): the Erdős-Straus equation 4/n = 1/a1 + 1/a2 + 1/a3 has at most O_eps(n^{3/5+eps}) solutions for every n.
• Corollary 2 (p. 2; proof pp. 10--11): an algorithm lists all representations of m/n as a sum of three unit fractions in expected time O_eps(n^eps (n^3/m^2)^{1/5}), and as a sum of k > 3 unit fractions in expected time O_{eps,k}(n^{2^{k-3}(8/5+eps)-1}). No result page.
• Theorem 2 (p. 2; proof pp. 14--16): f_4(m,n) <<_eps n^eps(n^{4/3}/m^{2/3} + n^{28/17}/m^{8/5}), and f_k(m,n) <<_eps (kn)^eps (k^{4/3}n^2/m)^{(28/17)·2^{k-5}} for k >= 5.
• Corollary 3 (p. 3): upper bounds for f_k(1,1) and for the number of solutions of 1 = sum 1/a_i + 1/prod a_i.
• Theorem 3 (pp. 3--4; proof pp. 16--18): lower bounds for f_3(m,n): with the constant log 6 for infinitely many n, with log 3 on a set of density one, and for m = 4 with log 6 on a set of density one.
• Corollary 4 (p. 4): the paper's restatement, via Dirichlet's theorem, of Elsholtz and Tao's (log p)^{0.549} bound in residue classes; recorded on the Theorem 4 page.
• Theorem 4 (p. 4; proof pp. 18--20): for every m and every reduced residue class e mod f, infinitely many primes p = e mod f with f_3(m,p) >>{f,m} exp((5 log 2/(12 lcm(m,f)) + o{f,m}(1)) log p/log log p).

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