Ha 2019 many solutions s unit equation 1

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proposition_2: Ha and Soundararajan's main technical estimate: for y >= 10 and integers 1 <= l <= k <= y^{1/3}/(log y)^2, the number N(y; k, l) of prime tuples with p_1...p_k = 1 mod q_1...q_l, p_i in (y/2, y] and q_j in (y/4, y/2], is lambda^l P^k (1 + O(1/log y)) when l <= k/2, with an added error O(l^{k-l} (4 lambda P)^l y^{k/2}) when k/4 <= l <= k/2.

theorem_1: Ha and Soundararajan's theorem that for every s some set S of s primes gives the equation a + 1 = c at least of order exp(s^{1/4}/log s) solutions with every prime factor of ac in S, improving the exponents 1/16 of Konyagin and Soundararajan and 1/6 - eps of Harper.

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Junsoo Ha, Kannan Soundararajan, Many solutions to the S-unit equation a + 1 = c. Acta Mathematica Hungarica 160 (2020), 153-160, doi:10.1007/s10474-019-00948-z. arXiv:1902.07397. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1902.07397), every other right reserved.

Theorem 1 shows that for all s there exist sets S of s primes for which the equation a + 1 = c has at least of order exp(s^{1/4}/log s) solutions with every prime factor of ac lying in S, improving Konyagin-Soundararajan's exp(s^{1/16}) and Harper's exp(s^{1/6-eps}) for this special case of the binary S-unit equation. The proof deduces Theorem 1 from Proposition 2, which, for integers 1 <= l <= k <= y^{1/3}/(log y)^2, evaluates N(y; k, l) asymptotically when l <= k/2 and, with an added error term, when k/4 <= l <= k/2 (the ranges as printed); N(y; k, l) counts the tuples of primes p_1, ..., p_k in (y/2, y] and q_1, ..., q_l in (y/4, y/2] with p_1...p_k congruent to 1 modulo q_1...q_l, and the paper views the result as an average statement on the equidistribution of smooth numbers in arithmetic progressions. The introduction surveys the surrounding landscape: Evertse's upper bound of 3 x 7^{2s+1} solutions for a + b = c, Erdős-Stewart-Tijdeman's exp((4-eps)(s/log s)^{1/2}) and Konyagin-Soundararajan's exp(s^{2-sqrt 2 -eps}) for the general equation, and, when S is the first s primes, Lagarias-Soundararajan's exp(s^{1/8-eps}) under GRH and Harper's unconditional exp(s^delta). Heuristics suggest exp(s^{1/2-eps}) solutions for a + 1 = c with S the first s primes and no more than exp(s^{1/2+eps}) in general. The paper does not mention problem 126; its results count consecutive S-units and give no bound for the number of distinct prime factors of the product of pairwise sums that the problem asks about.

Source: https://arxiv.org/abs/1902.07397.

Bears on. #126: the paper does not mention the problem; Theorem 1 (p. 2) counts solutions of a + 1 = c in S-units and gives no bound for the problem's product of pairwise sums.

Results. Pages are those of arXiv:1902.07397v1.

• Theorem 1 (p. 2): For all s there are sets S of s primes such that a + 1 = c has at least of order exp(s^{1/4}/log s) solutions with all prime factors of ac in S.
• Proposition 2 (pp. 2-3): For y >= 10 and integers 1 <= l <= k <= y^{1/3}/(log y)^2, N(y; k, l) = lambda^l P^k (1 + O(1/log y)) when l <= k/2, and the same plus O(l^{k-l} (4 lambda P)^l y^{k/2}) when k/4 <= l <= k/2 (the ranges as printed), where N(y; k, l) counts tuples of k primes in (y/2, y] and l primes in (y/4, y/2] whose first product is 1 mod the second, lambda is the sum of 1/q over primes q in (y/4, y/2] and P the number of primes in (y/2, y]; Theorem 1 is deduced from it.
• Evertse's bound (cited, pp. 1-2): The binary S-unit equation has at most 3 x 7^{2s+1} solutions; the authors know no better upper bound even for a + 1 = c.
• Heuristic (p. 2): For S the first s primes, a + 1 = c is expected to have exp(s^{1/2-eps}) solutions, with at most exp(s^{1/2+eps}) for general S.

The copy read for this card is the arXiv version, arXiv:1902.07397v1.

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