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On the representation of almost primes by pairs of quadratic forms. (English) Zbl 1146.11047
Let \(q_i(x, y)\) \((i= 1,2)\) be two integral binary quadratic forms which are non-singular and non-proportional. Suppose further that the product \(q_1q_2\) has no fixed prime factor, and that \(q_1(1, 0)\equiv q_2(1, 0)\equiv 1\pmod 4\). Then it is shown that \(q_1(x, y)q_2(x,y)\) is infinitely often a \(P_5\) almost-prime. The most difficult part of the proof is the establishment of a suitable “level-of-distribution” result. Here it is shown, very roughly speaking, that if \(x\) and \(y\) take values of size up to \(M\), say, then the product \(q_1(x, y)q_2(x, y)\) is well distributed for divisors up to \(M^{2-\delta}\), for any fixed \(\delta> 0\). The argument here is based in part on that of S. Daniel [J. Reine Angew. Math. 507, 107–129 (1999; Zbl 0913.11041)].
Once this is established the author uses a standard weighted sieve due to H. Diamond and H. Halberstam [Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 237, 101–107 (1997; Zbl 0941.11034)].

MSC:
11N36 Applications of sieve methods
11N05 Distribution of primes
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