Integers of the form \(n^ 2+1\) without large prime factors
A result of \textit{A. Schinzel} [Acta Arith. 13, 177-236 (1967; Zbl 0159.07101)] states that for every \(\alpha>0\) there exist infinitely many integers \(n\) such that \(n^2+1\) is free of prime factors \(>n^\alpha\). Schinzel's method, however, does not give a good lower bound for the counting function of such \(n\). Since the set of all integers \(n\) that are free of prime factors \(>n^\alpha\) has positive density for any \(\alpha>0\), one is led to conjecture that the same is true for the set of integers of the form \(n^2+1\). In the present paper, the author establishes this conjecture for exponents \(\alpha> 149/179\); that is, he shows that for such \(\alpha\) and all sufficiently large \(x\) one has \[ \# \bigl\{n \leq x: P^+ (n^2+1)< x^\alpha \bigr\} \gg x, \] where \(P^+ (n)\) denotes the greatest prime factor of \(n\). The proof of this result is an impressive tour de force, combining methods devised by \textit{C. Hooley} [Applications of sieve methods to the theory of numbers, Cambridge Tracts in Mathematics 70, Cambridge University Press, London (1976; Zbl 0327.10044)] and \textit{J.-M. Deshouillers} and \textit{H. Iwaniec} [Ann. Inst. Fourier 32, 1-11 (1982; Zbl 0509.10029)] for the dual problem of bounding \(P^+ (n^2+1)\) from below with weighted sieve techniques introduced by \textit{A. Balog} [Sémin. Théor. Nombres, Univ. Bordeaux I 1983/1984, Exp. No. 31 (1984; Zbl 0588.10052)] and \textit{J. Friedlander} [Number theory and applications, NATO ASI Ser., Ser. C 265, 393-401 (1989; Zbl 0686.10030)] to obtain upper bounds for the greatest prime factor of numbers of the form \(p+a\). (Same review submitted to MR).
upper bounds for the greatest prime factor
scientific article; zbMATH DE number 923731