Konyagin 2007 two s unit equations many solutions

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theorem_1: Konyagin and Soundararajan's construction, for every positive beta below 2 - sqrt 2, of arbitrarily large sets S of s primes for which the S-unit equation a + b = c has at least exp(s^beta) coprime solutions.

theorem_2: Konyagin and Soundararajan's construction of arbitrarily large sets S of s primes for which a + 1 = c has at least exp(s^{1/16}) solutions with every prime factor of ac in S, in the stronger form of arbitrarily large N with at least exp((log N)^{1/16}) divisors d such that d(d+1) divides N.

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Sergei Konyagin, Kannan Soundararajan, Two S-unit equations with many solutions. Journal of Number Theory 124 (2007), 193-199, doi:10.1016/j.jnt.2006.07.017. arXiv:math/0604453. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0604453), every other right reserved.

Konyagin and Soundararajan exhibit large prime sets S making two S-unit equations unusually rich in solutions. Theorem 1 shows that for any positive beta < 2 - sqrt(2) there are arbitrarily large sets S of s primes for which a + b = c has at least exp(s^beta) coprime solutions with all prime factors of abc in S, improving the exp((4-eps) sqrt(s/log s)) construction of Erdős, Stewart and Tijdeman. Theorem 2 treats the much more restrictive a + 1 = c and produces arbitrarily large S with at least exp(s^{1/16}) solutions, and in the stronger form arbitrarily large N with #{d : d(d+1) | N} >= exp((log N)^{1/16}), advancing a line of Erdős and Hall. The proof of Theorem 1 is a counting argument: squarefree numbers with prescribed numbers of prime factors in dyadic ranges, Cauchy-Schwarz on residue classes mod m to force many congruent pairs. The authors also guess that a + 1 = c has at most exp(s^{1/2+eps}) solutions, noting nothing substantially better than Evertse's exp(4s+6) upper bound is known. The paper does not mention problem 126; its Theorem 2 gives sets of at least exp(s^{1/16}) integers a for which the product of all a(a+1) has at most s distinct prime factors, and says nothing about the product of the sums of two distinct elements of one set that the problem asks about.

Source: https://arxiv.org/abs/math/0604453.

Bears on. #126: background only. The paper does not mention the problem; Theorem 2 (p. 1) gives sets of at least exp(s^{1/16}) integers a for which the product of all a(a+1) has at most s distinct prime factors, and says nothing about the product of the sums of two distinct elements of one set.

Results.

• Theorem 1 (p. 1): For any positive beta < 2 - sqrt(2) there exist arbitrarily large sets S of s primes such that a + b = c has at least exp(s^beta) coprime solutions with all prime factors of abc in S.
• Theorem 2 (p. 1): There exist arbitrarily large sets S of s primes with at least exp(s^{1/16}) solutions of a + 1 = c with all prime factors of ac in S; in the stronger form, arbitrarily large N with #{d : d(d+1) | N} >= exp((log N)^{1/16}). The page also records the authors' guess (p. 2) that for any set S of s primes, a + 1 = c has at most exp(s^{1/2+eps}) solutions.

The copy read for this card is the arXiv version, arXiv:math/0604453v1, and the result pages cite its page numbers (pp. 1-6).

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