Criteria for diophantine inequalities, indefinite principal forms and principal ideals of quadratic number fields (Q2522430)

This study is concerned with the ``diophantine'' inequalities \[ \vert q_0(x,y)\vert \equiv \vert (1, 0, -d/4)\vert \equiv \vert x^2 - dy^2/4\vert < \sqrt{d}/2, \tag{A} \] \[ \vert q_1(x,y)\vert \equiv \vert (1, 1, -(d-1)/4)\vert \equiv \vert x^2 + xy - (d-1)y^2/4\vert < \sqrt{d}/2, d>0, \tag{B} \] where \(d\) is the discriminant of the principal forms. The following theorems are proved: (1) Let \(D = d/4\) be a non-square integer, and \(x, y\) integers. Except for the trivial solution \(x y = 0\), \((x, y)\) is a proper solution of (A) if and only if \(\vert x\vert/\vert y\vert\) is an approximant of the diagonal of Minkowski continued fraction expansion (a diagonal approximant) for \(\sqrt{D}\). (2) Let \(D\equiv 1 \bmod 4\), and \(G = (D - 1)/4\) be a non-square integer. Then \((x, y)\), \(x > 0\), \(y > 0\), is a proper solution of \[ \vert x^2 + xy - Gy^2\vert < \sqrt{G} \] if and only if \(x/y\) is a diagonal approximant of \((-1 + \sqrt{D})/2\). (3) The diophantine inequality \[ \vert x^2 + xy - Gy^2\vert \le \sqrt{G}, \] where \(4G + 1\) is a non-square integer, has the proper solution \(x = a (> 0)\), \(y = b (>0)\), if and only if \(a/b\) is a diagonal approximant of the number \((-1 + \sqrt{4G + 1})/2\). The trivial exception is the case where \(G =g^2=\) a perfect square, \((g > 0)\), and then only if \(a =g - 1\), \(b = 1\), and \(a/b\) is an approximant, but not a diagonal approximant. (4) Consider the infinitely many representations \((x, y)\) of a given number of absolute value \(< \sqrt{D}/2\) by one of the indefinite quadratic forms \(q_n(x, y) = (1, n, -(d-n)/4)\), \(n = 0,1\). The necessary and sufficient condition that \((x, y)\), \(x\ne 0\), \(y\ne 0\), be a proper solution of the inequality \(\vert q_n(x, y) \vert < \sqrt{d}/2\) is that \(\vert x\vert/\vert y\vert\) be the diagonal approximant of \((\mp n+ \sqrt{d})/2\), \(xy\gtrless 0\). The trivial exception \(n = 1\), \(d =4g^2 + 1\), \(g> 0\) is also considered. (5) Further theorems deal with the representations \((x, y)\) of a number of absolute value less than \(\sqrt{d}/2\) by \(q_n(x, y)\) with \(d > 0\), and with the norms of algebraic numbers. Numerical illustrations are given.