Some fourth degree Diophantine equations in Gaussian integers.

*(English)*Zbl 1136.11304From the introduction: The solution \((x_0,y_0,z_0)\) of the equation \(ax^4+by^4=cz^2\) is called trivial if \(x_0=0\) or \(y_0=0\). P. Fermat showed that the equation \(x^4+y^4=z^2\) has
only trivial solutions in integers. D. Hilbert [Deutsche Math. Ver. 4, I–XVIII u. 175–546 (1897; JFM 28.0157.05)] extended this result by showing that the
equation \(x^4+y^4=z^2\) has only trivial solutions in a larger domain, namely in the integers of \(\mathbb Q(\sqrt{-1})\). In fact from his proof, it follows that the equation \(x^4-y^4=z^2\) also has only trivial solutions. J. T. Cross [Math. Mag. 66, No. 2, 105–108 (1993; Zbl 0796.11011)] gave a new proof for Hilbert’s result.

We consider the following eight equations \(x^4+my^4=z^2\), where \(m=\pm 2^n\), \(0\leq n\leq 3\). The equations \(x^4-2y^4=z^2\), \(y^4+8y^4=z^2\) have nontrivial solutions in integers as shown by the solutions \((3,2,7)\), \((1,1,3)\), respectively. We show that the remaining six equations have only trivial solutions in the integers of the quadratic field \(\mathbb Q(\sqrt{-1})\). The \(m=\pm 1\) case is covered by Hilbert’s result, so we will deal only with four cases. It is worthwhile to point out that the equation \(x^4+2y^4=z^2\) has nontrivial solution in \(\mathbb Z[\sqrt{\pm 2}]\), as the solution \((1,\sqrt{\pm 2},3)\) shows.

It is proved in [L. J. Mordell, “Diophantine equations”, London-New York: Academic Press (1969; Zbl 0188.34503)], among various similar results, that the equation \(x^4-py^4=z^2\) has only trivial solutions in integers, where \(p\) is a prime \(p\equiv\pm 3,-5\pmod{16}\). We show that the equations \(x^4-py^4=z^2\), \(x^4-p^3y^4=z^2\) have only trivial solutions in the Gaussian integers, where \(p\) is a prime \(p\equiv 3\pmod{8}\). We like to point out that the equations \(x^4+py^4=z^2\), \(x^4+p^2y^4=z^2\) have nontrivial integer solutions when \(p=3\) as shown by the solutions \((1,1,2)\), \((2,1,5)\), respectively.

See also the author’s previous papers related to this topic in Acta Math. Acad. Paedagog. Nyházi. (N.S.) 20, 1–10 (2004; Zbl 1059.11029) and Indian J. Pure Appl. Math. 30, No. 9, 857–861 (1999; Zbl 1125.11311).

We consider the following eight equations \(x^4+my^4=z^2\), where \(m=\pm 2^n\), \(0\leq n\leq 3\). The equations \(x^4-2y^4=z^2\), \(y^4+8y^4=z^2\) have nontrivial solutions in integers as shown by the solutions \((3,2,7)\), \((1,1,3)\), respectively. We show that the remaining six equations have only trivial solutions in the integers of the quadratic field \(\mathbb Q(\sqrt{-1})\). The \(m=\pm 1\) case is covered by Hilbert’s result, so we will deal only with four cases. It is worthwhile to point out that the equation \(x^4+2y^4=z^2\) has nontrivial solution in \(\mathbb Z[\sqrt{\pm 2}]\), as the solution \((1,\sqrt{\pm 2},3)\) shows.

It is proved in [L. J. Mordell, “Diophantine equations”, London-New York: Academic Press (1969; Zbl 0188.34503)], among various similar results, that the equation \(x^4-py^4=z^2\) has only trivial solutions in integers, where \(p\) is a prime \(p\equiv\pm 3,-5\pmod{16}\). We show that the equations \(x^4-py^4=z^2\), \(x^4-p^3y^4=z^2\) have only trivial solutions in the Gaussian integers, where \(p\) is a prime \(p\equiv 3\pmod{8}\). We like to point out that the equations \(x^4+py^4=z^2\), \(x^4+p^2y^4=z^2\) have nontrivial integer solutions when \(p=3\) as shown by the solutions \((1,1,2)\), \((2,1,5)\), respectively.

See also the author’s previous papers related to this topic in Acta Math. Acad. Paedagog. Nyházi. (N.S.) 20, 1–10 (2004; Zbl 1059.11029) and Indian J. Pure Appl. Math. 30, No. 9, 857–861 (1999; Zbl 1125.11311).

##### MSC:

11D25 | Cubic and quartic Diophantine equations |