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
j
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