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Greatest first prime factor of values of entire polynomials (with an appendix written with Jean-François Mestre). (Plus grand facteur premier de valeurs de polynômes aux entiers.) (French. English summary) Zbl 1379.11083
Let \(P^+(n)\) denote the largest prime factor of the integer \(n\). Using techniques borrowed from [D.R. Heath-Brown, Proc. Lond. Math. Soc. (3) 82, No. 3, 554–596 (2001; Zbl 1023.11048) ] and [C. Dartyge, Proc. Lond. Math. Soc. (3) 111, No. 1, 1–62 (2015; Zbl 1323.11069)], the author proves that for any even unitary irreducible quartic polynomial \(\Phi\) of Galois group \((\mathbb{Z}/2\mathbb{Z})^2\), there exists a positive constant \(c_\Phi\) such that the set of integers \(n\) satisfying \(P^+(\Phi(n))\geq n^{1+c_\Phi}\) has positive lower density. Such a result has been recently proved in [C. Dartyge, Proc. Lond. Math. Soc. (3) 111, No. 1, 1–62 (2015; Zbl 1323.11069)] for \(\Phi(n)=n^4-n^2+1\). An appendix written jointly with J.-F. Mestre shed some interesting lights on the required resultant computations.

MSC:
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
11L07 Estimates on exponential sums
11N36 Applications of sieve methods
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