Elsholtz 2013 counting number solutions erdos straus

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proposition_1_4: For A, B > 1 and a positive integer k at most a fixed power of AB, the sum of tau(kab^2 + 1) over a <= A and b <= B is O(AB log(A + B) log(1 + k)).

proposition_1_6: For every odd perfect square n the equation 4/n = 1/x + 1/y + 1/z has no Type I and no Type II solution, so f_I(n) = f_II(n) = 0.

proposition_1_7: Bounds the Type I and Type II solution counts pointwise by n^(3/5) and n^(2/5) up to O(1/log log n) in the exponent, so that every prime p has at most p^(3/5 + o(1)) solutions of 4/p = 1/x + 1/y + 1/z.

proposition_1_9: Essentially classifies the primitive residue classes in which 4/n is a sum of three unit fractions with polynomial denominators: the large primes of a Type I (Type II) solvable class lie in finitely many classes from four (three) listed families, each of them solvable by polynomials.

theorem_1_1: The Type I and Type II solution counts of 4/n = 1/x + 1/y + 1/z sum to order N log^3 N over n up to N and to order N log^2 N over primes (the Type I upper bound with an extra log log N), giving N log^2 N << sum over primes of f(p) << N log^2 N log log N.

theorem_1_11: For fixed m > k >= 3 the Type II solutions of m/n = 1/t_1 + ... + 1/t_k number >> N (log N)^(2^(k-1) - 1) summed over n <= N and

N (log N)^(2^(k-1) - 2) / log log N summed over primes p <= N.

theorem_1_8: Gives f(n) >= exp((log 3 + o(1)) log n / log log n) for infinitely many n, f(n) >> (log n)^0.549 for n in a set of density 1, and f(p) >> (log p)^0.549 for primes p in a set of relative density 1.

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Elsholtz, Christian and Tao, Terence, Counting the number of solutions to the Erdős-Straus equation on unit fractions. J. Aust. Math. Soc. 94 (2013), no. 1, 50--105.

For f(n) the number of positive-integer solutions (x,y,z) of 4/n = 1/x + 1/y + 1/z, the paper proves a range of average upper and lower bounds, notably that the sum of f(p) over primes p <= N lies between a constant times N log^2 N and a constant times N log^2 N log log N, so a typical prime admits only a small number of solutions. Theorem 1.1 gives the two-sided bounds N log^3 N << sum_{n<=N} f_I(n), sum_{n<=N} f_II(n) << N log^3 N for the counts f_I and f_II of Type I and Type II solutions, and over primes N log^2 N << sum_{p<=N} f_I(p) << N log^2 N log log N and N log^2 N << sum_{p<=N} f_II(p) << N log^2 N; with f(p) = 3 f_I(p) + 3 f_II(p) for odd primes these give the prime bound and, by Markov's inequality, bounds valid for all but an epsilon-density set of primes. The method converts solvability of Type I and Type II solutions into counting quadruples (for instance f_I(p) is, up to a factor of two, the number of (a,c,d,f) with 4acd = p + f, f dividing 4a^2 d + 1 and acd <= 3p/4), then applies Brun-Titchmarsh and Bombieri-Vinogradov together with a divisor-sum bound for sum over a <= A, b <= B of tau(k a b^2 + 1) adapted from an argument of Erdos; the hardest step is the upper bound on f_I. Remark 1.3 records that Heath-Brown obtains the stronger lower bound N log^6 N for sum f(n), so most solutions for composite n are of neither type. The paper counts solutions for Erdos problem 242, the Erdos-Straus conjecture that f(n) > 0 for all n >= 2, which it calls unresolved (p. 2); it proves the conjecture for no given n.

Source: https://arxiv.org/abs/1107.1010.

The copy read for this card is arXiv:1107.1010v6 (2 August 2015, 55 pages; its header carries a later compile date), whose arXiv comment says the statement and proof of Theorem 1.9 (Proposition 1.9 in the print) were corrected; the published version, J. Aust. Math. Soc. 94 (2013), no. 1, 50--105, DOI 10.1017/S1446788712000468 (the arXiv listing's journal reference), was not compared, so the page numbers here are the preprint's. Read status: claims checked. Theorem 1.1, the prime-average corollary of p. 5, Propositions 1.4, 1.6, 1.7 and 1.9, Theorem 1.8, Theorem 1.11 with Remark 1.12 and Corollary 1.13 were read clause by clause on the page images of pp. 4--10; no proof was read beyond its outline. Result pages: Theorem 1.1 (p. 4, with the corollary of p. 5), Proposition 1.4 (p. 6), Proposition 1.6 (p. 6), Proposition 1.7 (pp. 6--7), Theorem 1.8 (p. 7), Proposition 1.9 (p. 8) and Theorem 1.11 (pp. 9--10, with Corollary 1.13). The arXiv record names arXiv's non-exclusive distribution license (arXiv:1107.1010), every other right reserved.

Bears on. #242: the results count solutions of $4/n=1/x+1/y+1/z$ and prove existence for no given $n$. Theorem 1.1 and its corollary give $N\log^2N\ll\sum_{p\le N}f(p)\ll N\log^2N\log\log N$; Proposition 1.7 gives $f(p)\ll p^{3/5+O(1/\log\log p)}$ for every prime $p$; Theorem 1.8 gives $f(p)\gg(\log p)^{0.549}$ for primes in a set of relative density 1; Proposition 1.6 shows that odd perfect squares have no Type I or Type II solutions, which the paper reads as ruling out strategies such as a finite set of covering congruences, and Proposition 1.9 essentially classifies the primitive residue classes solvable by polynomial identities; Theorem 1.11 bounds from below, on average only, the Type II counts of $m/n$ as a sum of $k$ unit fractions with $m>k\ge3$, whose case $k=3$ is the setting of Schinzel's generalization in the problem's commentary. Proposition 1.4 is a divisor-sum estimate used for Theorem 1.1.

Results to transcribe.

• Theorem 1.1: Average value of Type I and Type II solution counts: sum_{n<=N} f_I(n) and sum_{n<=N} f_II(n) are both of order N log^3 N.
• Corollary (p. 5): N log^2 N << sum_{p<=N} f(p) << N log^2 N log log N, so a typical prime has few solutions to 4/p = 1/x+1/y+1/z.
• Remark 1.3: Heath-Brown's stronger lower bound sum_{n<=N} f(n) >> N log^6 N shows most solutions for composite n are neither Type I nor Type II.
• Proposition 1.4: Upper bound for sum over a<=A, b<=B of tau(k a b^2 + 1), obtained by an argument of Erdos and needed for the Type I upper bound.
• Proposition 1.6: f_I(n) = f_II(n) = 0 for every odd perfect square n.
• Proposition 1.7: f_I(n) << n^{3/5+O(1/log log n)}, f_II(n) << n^{2/5+O(1/log log n)}, hence f(p) << p^{3/5+O(1/log log p)}.
• Theorem 1.8: Lower bounds for f(n) for infinitely many n and on a set of density 1, and for f(p) on a set of primes of relative density 1.
• Proposition 1.9: The primitive residue classes solvable by polynomials.
• Theorem 1.11: Average lower bounds for the Type II counts of m/n as a sum of k unit fractions, m > k >= 3; Corollary 1.13 on generalized Cayley surfaces.

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