Bui 2024 problem erdos graham granville selfridge integral

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conjecture_1: Bui, Pratt and Zaharescu's conjecture that for each fixed c in (0, 1), every non-square n sufficiently large in terms of c has t_n >= (log n)^{1-c}.

theorem_1_1: Bui, Pratt and Zaharescu's theorem that for each fixed c in (0, 1] the proportion of n <= x with t_n <= n^c tends to the proportion with P^+(n) <= n^c, which is rho(1/c).

theorem_1_2: Bui, Pratt and Zaharescu's theorem that for fixed small eps > 0 and large x, at least x exp(-(3 sqrt 2/2 + eps) sqrt(log x log log x)) integers n <= x have t_n <= exp(sqrt((2 + eps) log n log log n)).

theorem_1_3: Bui, Pratt and Zaharescu's theorem that for fixed c in (0, 1) there are arbitrarily large J, at least J^{1-c} integers j_i in [1, J) and an integer x >= exp(c^2 (log J)^2/(5 log log J)) with x(x+J) prod (x+j_i) a square.

theorem_1_4: Bui, Pratt and Zaharescu's effective lower bound: every sufficiently large non-square n has t_n >> (log log n)^{6/5} (log log log n)^{-1/5}.

theorem_3_1: Bui, Pratt and Zaharescu's uniform comparison: for large x and c between (log log log x)^2/log log x and 1, the n <= x with t_n <= x^c and those with P^+(n) <= x^c differ in number by O(x/(c log x)).

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Bui, Hung M. and Pratt, Kyle and Zaharescu, Alexandru, A problem of Erdős-Graham-Granville-Selfridge on integral points on hyperelliptic curves. Math. Proc. Cambridge Philos. Soc. 176 (2024), no. 2, 309--323, DOI 10.1017/S0305004123000488. The copy read for this card is arXiv:2211.12467v1, dated 22 November 2022; the result labels and pages below are those of that version.

For each n, t_n is the least t such that n+1, ..., n+t contains a subset whose product with n is a square (t_n = 0 when n is a square). Theorem 1.1 shows that for every fixed c in (0,1] the proportion of n <= x with t_n <= n^c tends to the proportion with largest prime factor P^+(n) <= n^c, which Remark 1 identifies as the Dickman-de Bruijn value rho(1/c); so for every fixed c > 0 a positive proportion of n have t_n <= n^c, against Granville's remark that presumably t_n > n^c for some fixed c > 0. Theorem 1.1 is deduced from the uniform Theorem 3.1, which compares the counts with thresholds x^c. Theorem 1.2 gives, for fixed small eps > 0 and x large depending on eps, at least x exp(-(3 sqrt(2)/2 + eps) sqrt(log x log log x)) integers n <= x with t_n <= exp(sqrt((2+eps) log n log log n)), far below any power of n; a modification of its proof gives Theorem 1.3, hyperelliptic curves of large genus with integral points of large height. In the other direction Theorem 1.4 gives an effective lower bound t_n >> (log log n)^{6/5} (log log log n)^{-1/5} for large non-square n, from height bounds for integral points on hyperelliptic curves, and Conjecture 1 proposes t_n >= (log n)^{1-c}. The method relates t_n to P^+(n), starting from Granville and Selfridge's result that t_n = P^+(n) when P^+(n) > sqrt(2n)+1, and uses smooth-number counts and an effective theorem of Bérczes, Evertse and Győry.

Source: https://arxiv.org/abs/2211.12467. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2211.12467), every other right reserved.

Read status: claims checked for Theorems 1.1--1.4 (p. 2), Theorem 3.1 (p. 3) and Conjecture 1 (p. 20), read clause by clause on the page images of the arXiv v1 edition; the proofs were read for structure only. Nothing here is independently reviewed.

Bears on. #841, which asks for estimates of $t_n$: the paper studies $t_n$ directly and says (abstract, p. 1) that it solves Granville's problem on the size of $t_n$ unconditionally. Theorem 1.1 gives the limiting distribution of $t_n$ on the scale $n^c$, Theorem 1.2 many $n$ with very small $t_n$, and Theorem 1.4 a lower bound for every large non-square $n$; none determines $t_n$ for an individual $n$. #437, on how many partial products of an increasing sequence in $\{1,\ldots,x\}$ can be squares: the paper does not mention partial products or the problem; its Theorem 1.2 is the input to Tao's later deduction, which is not in the paper.

Results.

• Theorem 1.1 (p. 2): For fixed c in (0,1], the limiting proportion of n <= x with t_n <= n^c equals that with P^+(n) <= n^c, namely rho(1/c).
• Theorem 1.2 (p. 2): For fixed sufficiently small eps > 0 and x large depending on eps, at least x exp(-(3 sqrt(2)/2 + eps) sqrt(log x log log x)) integers n <= x satisfy t_n <= exp(sqrt((2+eps) log n log log n)).
• Theorem 1.3 (p. 2): For fixed c in (0,1), there are arbitrarily large J with N >= J^{1-c} integers 1 <= j_1 < ... < j_N < J and a positive integer x >= exp(c^2 (log J)^2 / (5 log log J)) making x(x+J) prod_{i<=N} (x+j_i) a square.
• Theorem 1.4 (p. 2): For sufficiently large non-square n, t_n >> (log log n)^{6/5} (log log log n)^{-1/5}, with effectively computable implied constant.
• Theorem 3.1 (p. 3): For large x and (log log log x)^2/log log x <= c <= 1, the counts of n <= x with t_n <= x^c and with P^+(n) <= x^c differ by O(x/(c log x)), uniformly in c.
• Conjecture 1 (p. 20): For fixed c in (0,1), every non-square n sufficiently large in terms of c has t_n >= (log n)^{1-c}.

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