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The Diophantine equation \( x^4 + 2 y^4 = z^4 + 4 w^4\). (English) Zbl 1138.11056
In the “Workshop on Rational and Integral Points on Higher-Dimensional Varieties” held in Palo Alto CA (2002), Sir P. Swinnerton-Dyer posed the following problem: “Does there exist a \(K3\) surface \(S\) over \(\mathbb{Q}\) such that \(0<\# S(\mathbb{Q})< \infty\)?” [Problem/Questions 6a; Boston: Birkhäuser Prog. Math. 226, 235–257 (2004; Zbl 1211.11077)]. One possible candidate for a \(K3\) surface with the above property is the projective surface defined by the equation \(x^4+ 2y^4= z^4+ 4w^4\).
It has the \(\mathbb{Q}\)-rational points (1:0:1:0) and (1:0:\(-1\):0). Sir P. Swinnerton-Dyer posed also the problem to find a third rational point on this surface [Problem/Questions 6c (loc. cit.)].
The paper under review gives an answer to this problem. More precisely, a systematic search by computer, shows that the projective surface defined by \(x^4+ 2y^4= z^4+ 4w^4\) admits precisely ten \(\mathbb{Q}\)-rational points which allow integral coordinates within the hypercube \(|x|,|y|,|z|,|w|< 2,5\times 10^6\). These are the points \((\pm 1\):0:\(\pm 1\):0), \((\pm 1484801\):\(\pm 1203120\):\(\pm 1169407\):\(\pm 1157520)\).

MSC:
11Y50 Computer solution of Diophantine equations
11D25 Cubic and quartic Diophantine equations
14G05 Rational points
Software:
ARIBAS
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