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The largest prime factor of \(X^3+2\). (English) Zbl 1384.11093
It was shown by the reviewer [Proc. Lond. Math. Soc. (3) 82, No. 3, 554–596 (2001; Zbl 1023.11048)] that there is a positive constant \(\varpi\) such that one has \[ P(n^3+2)>n^{1+\varpi} \] for a positive proportion of integers \(n\) in any sufficiently long range \((X,2X]\). Here \(P(m)\) denotes the largest prime factor of the integer \(m\). The reviewer’s proof produces \(\varpi=10^{-303}\), which is disappointingly small.
The present paper shows how to improve one part of the reviewer’s argument, and allows one to take \(\varpi=10^{-52}\). Unfortunately this is still rather small.
The paper focuses on the quantity \(T(h,\delta;X)\), which is the number of integers \(n\in(X,2X]\) for which \(n^3+2\) has at least \(h\) prime factors \(p>X^{\delta}\). Roughly speaking, the paper shows that \(T(h,\delta;X)\leq 2^{-h}X\) for the range of parameters under consideration, provided that \(h\) is larger than a certain multiple of \(\log(1/\delta)\). In effect this provides a bound for the key quantity \[ \sum_h \min(h,\delta^{-1})T(h,\delta;X) \] which one may think of as being polynomial in \(\delta\), rather than exponential as in the reviewer’s work. However one eventually works with \(\delta=1/321\), so that such an asymptotic interpretation needs to be treated with caution.
The paper is necessarily quite technical, but would be a good starting point for anyone hoping to obtain a “realistic” value for \(\varpi\).
MSC:
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11N25 Distribution of integers with specified multiplicative constraints
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[1] [1]P. Billingsley, On the distribution of large prime divisors, Period. Math. Hungar. 2 (1972), 283–289. · Zbl 0242.10033
[2] [2]R. de la Bret‘eche, Plus grand facteur premier de valeurs de polyn\hat{}omes aux entiers, Acta Arith. 169 (2015), 221–250. · Zbl 1379.11083
[3] [3]C. Dartyge, Le probl‘eme de Tch\'{}ebychev pour le douzi‘eme polyn\hat{}ome cyclotomique, Proc. London Math. Soc. (3) 111 (2015), 1–62. 80 · Zbl 1323.11069
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