zbMATH — the first resource for mathematics

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), \((\pm 1484801\):\(\pm 1203120\):\(\pm 1169407\):\(\pm 1157520)\).

11Y50 Computer solution of Diophantine equations
11D25 Cubic and quartic Diophantine equations
14G05 Rational points
Full Text: DOI
[1] Arnaud Beauville, Complex algebraic surfaces, London Mathematical Society Lecture Note Series, vol. 68, Cambridge University Press, Cambridge, 1983. Translated from the French by R. Barlow, N. I. Shepherd-Barron and M. Reid. · Zbl 0512.14020
[2] Daniel J. Bernstein, Enumerating solutions to \?(\?)+\?(\?)=\?(\?)+\?(\?), Math. Comp. 70 (2001), no. 233, 389 – 394. · Zbl 0960.11055
[3] Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest, Introduction to algorithms, The MIT Electrical Engineering and Computer Science Series, MIT Press, Cambridge, MA; McGraw-Hill Book Co., New York, 1990. · Zbl 1158.68538
[4] Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273 – 307 (French). · Zbl 0287.14001
[5] Forster, O.: Algorithmische Zahlentheorie, Vieweg, Braunschweig 1996.
[6] Bjorn Poonen and Yuri Tschinkel , Arithmetic of higher-dimensional algebraic varieties, Progress in Mathematics, vol. 226, Birkhäuser Boston, Inc., Boston, MA, 2004. · Zbl 1054.11006
[7] Robert Sedgewick, Algorithms, Addison-Wesley Series in Computer Science, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. · Zbl 0529.68002
[8] Nigel P. Smart, The algorithmic resolution of Diophantine equations, London Mathematical Society Student Texts, vol. 41, Cambridge University Press, Cambridge, 1998. · Zbl 0907.11001
[9] Swinnerton-Dyer, Sir P.: Rational points on fibered surfaces, in: Tschinkel, Y. : Mathematisches Institut, Seminars 2004, Universitätsverlag, Göttingen 2004, 103-109. · Zbl 1101.14027
[10] André Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Actualités Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann et Cie., Paris, 1948 (French). · Zbl 0036.16001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.