an:05015502 Zbl 1138.11056 Elsenhans, Andreas-Stephan; Jahnel, J??rg The Diophantine equation $$x^4 + 2 y^4 = z^4 + 4 w^4$$ EN Math. Comput. 75, No. 254, 935-940 (2006). 00124338 2006
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11Y50 11D25 14G05 $$K3$$ surface; diagonal quartic surface; rational point; Diophantine equation; computer solution In the Workshop on Rational and Integral Points on Higher-Dimensional Varieties'' held in Palo Alto CA (2002), Sir \textit{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: Birkh??user 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), $$(\pm 1484801$$:$$\pm 1203120$$:$$\pm 1169407$$:$$\pm 1157520)$$. D. Poulakis (Thessaloniki) Zbl 1101.14027; Zbl 1211.11077