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Chebyshev’s problem for the twelfth cyclotomic polynomial. (Le problème de Tchébychev pour le douzième polynôme cyclotomique.) (French. English summary) Zbl 1323.11069
It was shown by Chebyshev that there are infinitely many \(n\) for which the largest prime factor of \(n^2+1\) is strictly larger than \(n\). This was strengthened by C. Hooley [Acta Math. 117, 281–299 (1967; Zbl 0146.05704)], who showed that the largest prime factor is at least \(n^{11/10}\) infinitely often. The reviewer [Proc. Lond. Math. Soc. (3) 82, No. 3, 554–596 (2001; Zbl 1023.11048)] proved a corresponding result for the largest prime factor \(p\) of \(n^3+2\), showing that there is a constant \(\delta>0\) such that \(p>n^{1+\delta}\) for infinitely many \(n\). Indeed one may take \(\delta=10^{-303}\). The papers by Hooley and the reviewer use incomplete exponential sums. In Hooley’s case the length of the exponential sum is comparable with its modulus, but for \(n^3+2\) the length is around the square-root of the modulus. However, one is able to arrange that the modulus factors sufficiently well for the \(q\)-analogue of van der Corput’s method to be used.
The present paper handles a quartic polynomial for the first time. It is shown that there are infinitely many \(n\) for which the largest prime factor of \(n^4-n^2+1\) is at least \(n^{1+\delta}\), where one can take \(\delta=10^{-26531}\). The proof follows the same basic line of attack as in the reviewer’s work. However there is a major obstacle at the very start, when one expresses the problem in terms of exponential sums. The reviewer’s procedure is somewhat ad hoc and it is by no means clear how to extend it to higher degree polynomials, or indeed whether such a generalization exists. Thus it is the corresponding argument in the present paper which is the most important forward step. The author makes heavy use of the explicit shape of the polynomial \(X^4-X^2+1\), and it is not at all clear to what extent it will generalize to other quartic polynomials.
In addition to the exponential sums which occur it is necessary to handle sieving conditions involving lattices in \(\mathbb{Z}^3\). The paper provides a level-of-distribution result in this context, somewhat in the spirit of S. Daniel [J. Reine Angew. Math. 507, 107–129 (1999; Zbl 0913.11041)] and G. Marasingha [J. Lond. Math. Soc., II. Ser. 82, No. 2, 295–316 (2010; Zbl 1269.11099)]. The sieve argument can then be completed along the lines of the reviewer’s paper.

MSC:
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11N36 Applications of sieve methods
11P21 Lattice points in specified regions
11L07 Estimates on exponential sums
11L26 Sums over arbitrary intervals
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