On the representation of integers by binary forms.

*(English)*Zbl 0102.03601The most striking result is the following consequence of Theorem 2. Let \(F (x, y)\) be a binary form of degree \(n\) with rational integral coefficients and nonzero discriminant such that \(F(1, 0)\neq 0\), \(F(0, 1) \neq 0\). Let a be the height of \(F\) (= maximum of the absolute value of the coefficients and let \(p_1,\dots, p_t\) be a fixed set of \(t\) primes.
Then

(i) there are at most \(c_1 (a n)^{c_2n^{1/2}}+c_3n^{t+1}\) coprime pairs of integers \(x, y\) for which \(F (x, y)\) has no prime factors distinct from \(p_1,\dots, p_t\);

(ii) there are at most \((c_3 n)^{t+1}\) coprime pairs of integers \(x, y\) such that \(F(x,y)\) is greater than a certain constant depending an \(a\) and \(n\) and for which \(F(x,y)\) has no prime factors distinct from \(p_1,\dots, p_t\).

(iii) Let \(p\) be a prime. Then if \(p\) is sufficiently large there are at most \(c_4 n^2\) coprime pairs of integers for which \(\pm F(x,y)\) is equal to \(p\) or a power of \(p\). Here \(c_1,c_2,c_3,c_4\) are constants which can be given explicitly and are not too large.

Theorem 2 itself gives better but very elaborate estimates.

Statement (i) above gives, in particular, the upper bound \(c_1 (a n)^{c_2n^{1/2}}+c_3n^{t+1}\) for the number of solutions of \(F (x, y) = m\), where \(t\) is the number of prime factors of \(m\), and this is much better than the estimate obtained by H. Davenporth and K. F. Roth [Mathematika 2, 160–167 (1955; Zbl 0066.29302)]. The proof is an elaboration of an earlier one of K. Mahler [Math. Ann. 107, 691–730 (1933; Zbl 0006.10502; JFM 59.0220.01)] of a less precise result (which as the authors point out is, however in general much stronger than that of Davenport and Roth), and is related closely to Mahler’s work an the \(p\)-adic Thue–Siegel theorem. Of independent interest are Lemma 1 giving a lower bound for \(f (z)\), where \(f\) is a polynomial wich complex coefficients, in terms of the height, discriminant and degree of \(f\) and of the minimum distance of \(z\) from a zero of \(f\) which is stronger than that given by N. I. Fel’dman [Izv. Akad. Nauk SSSR, Ser. Mat. 15, 53–74 (1951; Zbl 0042.04802)] and the \(p\)-adic analogue (Lemma 2) which improves an estimate of F. Kasch and B. Volkmann [Math. Z. 72, 367–378 (1960; Zbl 0091.04801)].

Using these the authors obtain a lower bound for \[ |F (x, y)|\prod_{\tau=1}^t |F(x,y)|_{p_{\tau}} \] in terms of the minimum distance of \(x/y\) from the complex and \(p\)-adic zeros of \(F (x, 1) = 0\). When the only prime factors of \(F (x, y)\) are \(p_1,\dots, p_t\) then the expression (*) is unity. Hence the machinery of the \(p\)-adic Thue–Siegel theorem applies. There is an elaborate and ingenious subdivision into cases. There is also a rather elaborate generalization of Theorem 2 (Theorem 3) in which, roughly speaking, although \(F (x, y)\) is not entirely composed of \(p_1,\dots, p_t\), the product of the primes dividing \(F(x, y)\) other than these is of lower order of magnitude.

(i) there are at most \(c_1 (a n)^{c_2n^{1/2}}+c_3n^{t+1}\) coprime pairs of integers \(x, y\) for which \(F (x, y)\) has no prime factors distinct from \(p_1,\dots, p_t\);

(ii) there are at most \((c_3 n)^{t+1}\) coprime pairs of integers \(x, y\) such that \(F(x,y)\) is greater than a certain constant depending an \(a\) and \(n\) and for which \(F(x,y)\) has no prime factors distinct from \(p_1,\dots, p_t\).

(iii) Let \(p\) be a prime. Then if \(p\) is sufficiently large there are at most \(c_4 n^2\) coprime pairs of integers for which \(\pm F(x,y)\) is equal to \(p\) or a power of \(p\). Here \(c_1,c_2,c_3,c_4\) are constants which can be given explicitly and are not too large.

Theorem 2 itself gives better but very elaborate estimates.

Statement (i) above gives, in particular, the upper bound \(c_1 (a n)^{c_2n^{1/2}}+c_3n^{t+1}\) for the number of solutions of \(F (x, y) = m\), where \(t\) is the number of prime factors of \(m\), and this is much better than the estimate obtained by H. Davenporth and K. F. Roth [Mathematika 2, 160–167 (1955; Zbl 0066.29302)]. The proof is an elaboration of an earlier one of K. Mahler [Math. Ann. 107, 691–730 (1933; Zbl 0006.10502; JFM 59.0220.01)] of a less precise result (which as the authors point out is, however in general much stronger than that of Davenport and Roth), and is related closely to Mahler’s work an the \(p\)-adic Thue–Siegel theorem. Of independent interest are Lemma 1 giving a lower bound for \(f (z)\), where \(f\) is a polynomial wich complex coefficients, in terms of the height, discriminant and degree of \(f\) and of the minimum distance of \(z\) from a zero of \(f\) which is stronger than that given by N. I. Fel’dman [Izv. Akad. Nauk SSSR, Ser. Mat. 15, 53–74 (1951; Zbl 0042.04802)] and the \(p\)-adic analogue (Lemma 2) which improves an estimate of F. Kasch and B. Volkmann [Math. Z. 72, 367–378 (1960; Zbl 0091.04801)].

Using these the authors obtain a lower bound for \[ |F (x, y)|\prod_{\tau=1}^t |F(x,y)|_{p_{\tau}} \] in terms of the minimum distance of \(x/y\) from the complex and \(p\)-adic zeros of \(F (x, 1) = 0\). When the only prime factors of \(F (x, y)\) are \(p_1,\dots, p_t\) then the expression (*) is unity. Hence the machinery of the \(p\)-adic Thue–Siegel theorem applies. There is an elaborate and ingenious subdivision into cases. There is also a rather elaborate generalization of Theorem 2 (Theorem 3) in which, roughly speaking, although \(F (x, y)\) is not entirely composed of \(p_1,\dots, p_t\), the product of the primes dividing \(F(x, y)\) other than these is of lower order of magnitude.

Reviewer: J. W. S. Cassels

##### MSC:

11E16 | General binary quadratic forms |