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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
ARIBAS
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##### References:
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