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Some fourth degree Diophantine equations in Gaussian integers. (English) Zbl 1136.11304
From 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).
MSC:
 11D25 Cubic and quartic Diophantine equations
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