Chan 2015 factors almost squares lattice points circles

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theorem_2: Chan's theorem that every sufficiently large n = (N-a)(N+b) with 0 <= a <= b <= exp((log n)^{2/7}) has at most eighteen divisors within n^{1/4}(log n)^{1/14} of sqrt(n).

theorem_3: Chan's theorem that for every sufficiently large perfect square n at most ten integer points (a, b) with a^2 + b^2 = n have |b| < n^{1/4}(log n)^{1/7}.

theorem_4: Chan's theorem that for sufficiently large n with n = a_1^2 + b_1^2 and |b_1| <= exp((log n)^{2/7}), at most thirty-six integer points (a, b) with a^2 + b^2 = n have |b| < n^{1/4}(log n)^{1/14}.

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Chan, Tsz Ho, Factors of almost squares and lattice points on circles. Int. J. Number Theory 11 (2015), no. 5, 1701--1708. https://doi.org/10.1142/S1793042115400205

Continuing the author's earlier work on the Erdos-Rosenfeld problem (Conjecture 1 here) and Ruzsa's stronger form (Conjecture 2), Theorem 2 shows that any sufficiently large n that factors as (N-a)(N+b) with 0 <= a <= b <= exp((log n)^{2/7}) - the almost squares, including N^2-1, N^2-4, N^2-N-6 - has at most eighteen divisors within n^{1/4}(log n)^{1/14} of sqrt(n), extending Theorem 1 (the perfect-square case with at most five divisors). Interpreting divisors as lattice points on the hyperbola xy = n, the same method is applied to the circle x^2 + y^2 = n and the conjecture (Conjecture 3) that for each alpha < 1/2 boundedly many lattice points have |b| in any window [N, N + n^alpha), in its special case N = 0 near the x-axis: Theorem 3 gives at most ten lattice points with |b| < n^{1/4}(log n)^{1/7} on x^2+y^2 = N^2 for large squares, and Theorem 4 gives, for large n, at most thirty-six such points with |b| < n^{1/4}(log n)^{1/14} when n = a_1^2 + b_1^2 with |b_1| <= exp((log n)^{2/7}). The main tool is Turk's quantitative bound (Theorem 5) on solutions of simultaneous Pell equations ax^2 - by^2 = e, cx^2 - dz^2 = f, together with its consequence (Theorem 6) on three integers of the form a_i x_i^2 in a short interval. The paper bears on problem 887, extending the class of n for which boundedly many divisors near sqrt(n) is proved.

Source: https://arxiv.org/abs/1406.2230. The copy read for this card is arXiv:1406.2230v1 (9 June 2014), 6 pages. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1406.2230), every other right reserved.

Bears on.

• Problem 887: Theorem 2 answers the question with K = 18 for every n = (N-a)(N+b) with 0 <= a <= b <= exp((log n)^{2/7}), since for each C > 0 the window C n^{1/4} lies inside n^{1/4}(log n)^{1/14} once n is large; it says nothing about other n.
• Problem 886: the abstract says the paper considers Ruzsa's conjecture (its Conjecture 2) for almost squares, but the window of Theorem 2 is shorter than n^{1/2-eps} for every eps < 1/4 once n is large, and for eps >= 1/4 it covers only almost squares, where the problem is already settled for every n; it settles no instance.

Results. Labels and pages are those of arXiv:1406.2230v1.

• Theorem 2 (p. 1): any sufficiently large n = (N-a)(N+b) with integers 0 <= a <= b <= exp((log n)^{2/7}) has at most eighteen divisors between sqrt(n) - n^{1/4}(log n)^{1/14} and sqrt(n) + n^{1/4}(log n)^{1/14}.
• Theorem 3 (p. 2): for sufficiently large perfect squares n = N^2, at most ten integer points (a, b) with a^2 + b^2 = n have |b| < n^{1/4}(log n)^{1/7}.
• Theorem 4 (p. 2): for sufficiently large n, if n = a_1^2 + b_1^2 for some |b_1| <= exp((log n)^{2/7}), then at most thirty-six integer points (a, b) with a^2 + b^2 = n have |b| < n^{1/4}(log n)^{1/14}.

Recalled, with no page here: Theorem 1 (p. 1), that any sufficiently large perfect square n = N^2 has at most five divisors between sqrt(n) - n^{1/4}(log n)^{1/7} and sqrt(n) + n^{1/4}(log n)^{1/7}, proved in Chan 2014 (Corollary 1.4 there); and two results of Turk used as tools (p. 2). Theorem 5: let a, b, c, d be squarefree positive integers with a != b and c != d, let e, f be integers, and if af = ce assume also that abcd is not a perfect square; then every positive integer solution of ax^2 - by^2 = e, cx^2 - dz^2 = f satisfies max(x,y,z) < exp(C alpha^2 (log alpha)^3 gamma log gamma), where alpha = max(a,b,c,d), beta = max(|e|,|f|,3), gamma = max(alpha log alpha, log beta) and C is a large absolute constant. Theorem 6: if [N, N+K] with K >= 3 contains three distinct integers a_i x_i^2 with positive integers a_i, x_i and H = max(a_1,a_2,a_3,3), then C H^2 (log H)^3 (H log H + log K)(log H + log log K) > log N for some absolute constant C.

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