A new characterization of the integer 5906
Tóm tắt
It is shown that 5906 is the. least integer expressible as the sum of two rational fourth powers but not as the sum of two integer fourth powers. The relevant Diophantine equation x4+y4=D represents a curve of genus 3, and extensive arithmetic calculations are involved: in particular, class-number, units and ideal-class stucture are. computed for four particular eighth degree extension fields of the rationals. The result provides several examples of curves of genus 3, everywhere locally solvable, but with no rational points.
Tài liệu tham khảo
E. Artin, Abh. Math. Sem. Hansische Univ3 (1924), 89;8 (1931), 292
Z.I. Borevic and I.R. Šafaveric, Number Theory, Academic Press, 1966
R. Brauer, Beziehungen Zwischen Klassenzahlen von Teilkörpern eines Galoisschen Körpers, Math. Nachr., 4 (1950), 158–174
J.W.S. Cassels and M.J.T. Guy, On the Hasse Principle for Cubic Surfaces Mathematika, 13, (1966), 111–120.
J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39, 1977, 223–251
H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, I, Jber. Deutschen Math.Ver.(1926)
D. Hubert, Die Theorie der algebraischen Zahlkö'rper, Gesammelte Abhandlungen Vol I, 63–363, Chelsea, New York, 1965
C.G.J. Jacobi, Canon Arithmeticus (nach Berechnungen von Patz neu herausgegeben von H. Brandt), Akademie-Verlag 1956
R.B. Lakein, Class Numbers of 5000 Quartic Fields\(\not Q/\sqrt \pi \), SUNY at Buffalo 1973 (Mathematics of Computation UMT file)
P. Morton, On Rédei's Theory of the Pell Equation, J reine angew. Math, 307, 1979, 373–398
E.S. Selmer, Tables for the Purely Cubic Field\(K(\sqrt[3]{m})\), Avh. Norske Vid, Akad. Oslo I, 1955 no.5
A. Weil, Sur les Courbes Algébriques et les Variétés qui s'en déduisent, Hermann, Paris, 1948.
D. Coray, Algebraic points on cubic hypersurfaces, Acta Arith. XXX, 1976, 267–296