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Friable values of binary forms. (English) Zbl 1268.11136
Let \(P^+(n)\) denote the largest prime factor of \(n\) and let \(F\in\mathbb Z[X,Y]\) be an integral binary form of degree \(t\geq 2\). The authors consider the problem of how small \(y\) can be as a function of \(x\) in order that \[ \Psi_F(x,y):= \text{card}\{1\leq a,b\leq x: P^+(F(a,b))\leq y\}\times x^2.\tag{\(*\)} \] When \(F= X^2+ Y^2\), P. Moree in [Manuscr. Math. 80, No. 2, 199–211 (1993; Zbl 0791.11046)] showed that \((*)\) holds with \(y= x^\varepsilon\). Suppose that \(F\in\mathbb Z[X,Y]\) with degree \(t\geq 2\) has no repeated irreducible factors, where \(k\) of these irreducible factors have the largest degree \(g\) and \(l\) have degree \(g-1\).
In Theorem 1 the authors prove that for any \(\varepsilon> 0\) \((*)\) holds for \(y\geq x^{\alpha_F+\varepsilon}\), where \(\alpha_F= g-{2\over k}\) if \(k\geq 2\) and, if \(k= 1\), \(\alpha_F= g-1-{1\over l+1}\) or \({2\over 3}\) according as \((g, t)\neq(2,3)\) or \((g, t)= (2, 3)\). When \(F\) is a cubic form Theorem 2 provides an improvement of this result by establishing that \(\alpha_F={1\over \sqrt{e}}\) or \(0\) according as \(F\) is irreducible or reducible. The proofs are intricate and depend on establishing three further propositions as well as utilizing results or methods contained in papers in the long list of references. In particular the proof of Theorem 1 extends the ideas in [Period. Math. Hung. 43, No. 1–2, 111–119 (2001; Zbl 0980.11041)] by C. Dartyge, G. Martin and G. Tenenbaum, where a similar problem for \(F\in\mathbb Z]X]\) was considered.

MSC:
11N25 Distribution of integers with specified multiplicative constraints
11N36 Applications of sieve methods
11E76 Forms of degree higher than two
11Y05 Factorization
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