Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T23:51:09.217Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Diophantine equations with almost all solutions trivial

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

George Greaves
George Greaves
Affiliation:
School of Mathematics, University of Wales, Cardiff, 23 Senghennydd Road, P.O. Box 926, Cardiff CF2 4YH
Get access

Extract

In this paper we investigate the solutions in integers x, y, z, X, Y, Z of the system

where P is a real number that may be taken to be arbitrarily large, and the (fixed) integer exponent h satisfies h≥4. The system has 6P3 + O(P2) “trivial” solutions in which x, y, z are a permutation of X, Y, Z. Our result implies that the number of non-trivial solutions is at most o(P3), so that the total number of solutions is asymptotic to 6P3.

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation
Type
Research Article
Information
Mathematika , Volume 44 , Issue 1 , June 1997 , pp. 14 - 36
Copyright
Copyright © University College London 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Boklan, K. D.. The asymptotic formula in Waring's problem. Mathematika, 41 (1994). 329–347.CrossRefGoogle Scholar
2.Bombieri, E. and Pila, J.. The number of integral points on arcs and ovals. Duke Math. J., 59 (1989), 337–357.CrossRefGoogle Scholar
3.Bremner, A.. A geometric approach to equal sums of fifth powers. J. Number Theory, 13 (1981), 337–354.CrossRefGoogle Scholar
4.Cassels, J. W. S.. An Introduction to the Geometry of Numbers (Springer, Berlin, 1959).Google Scholar
5.Dickson, L. E.. History of the Theory of Numbers, Vol. I (Carnegie, Washington, 1919).Google Scholar
6.Erdős, P.. On the integers of the form xk + yk J. London Math. Soc., 14 (1939), 250–254.CrossRefGoogle Scholar
7.Gallagher, P. X.. The large sieve and probabilistic Galois theory. Proc. Sympos. Pure Math., 24, (American Math. Soc, Providence, 1973), pp. 91–101.Google Scholar
8.Greaves, G.. Power-free values of binary forms. Quarterly J. Math. Oxford (2), 43, (1992), 45–65.CrossRefGoogle Scholar
9.Greaves, G.. On the representation of a number as a sum of two fourth powers (in Russian). Math. Zametki, 55 (1993), 47–58. English translation in Math. Notes, 55 (1994), 134–141.Google Scholar
10.Hooley, C.. On another sieve method and the numbers that are a sum of two h-th powers. Proc. London Math. Soc., 43 (1981), 73–109.CrossRefGoogle Scholar
11.Hooley, C.. On another sieve method and the numbers that are a sum of two h-th powers: II. J. Reine Angew. Math., 475 (1996), 55–75.Google Scholar
12.Hua, L.-K.. Additive Theory of Prime Numbers (American Math. Soc., Providence, 1965).Google Scholar
13.Hua, L.-K.. Introduction to Number Theory (Springer, Berlin, 1982).Google Scholar
14.Huxley, M. N.. The large sieve inequality for algebraic number fields. Mathematika, 15 (1968), 178–187.CrossRefGoogle Scholar
15.Jarnik, V.. Über die gitterpunkte auf konvexen kurven. Math. Zeitschr., 24 (1925), 500–518.CrossRefGoogle Scholar
16.Lander, L. J., Parkin, T. R. and Selfridge, J. L.. A survey of equal sums of like powers. Maths. of Computation, 21 (1967), 446–459.CrossRefGoogle Scholar
17.Montgomery, H. L.. Topics in multiplicative number theory. Lecture Notes in Mathematics, 227 (Springer, Berlin, 1971).Google Scholar
18.Schmidt, W. M.. Equations over finite fields. Lecture Notes in Mathematics, 536 (Springer, Berlin, 1976).Google Scholar
19.Skinner, C. M. and Wooley, T. D.. Sums of two k-th powers. J. Reine Angew. Math., 462 (1995), 57–68.Google Scholar
20.Skinner, C. M. and Wooley, T. D.. On the paucity of non-diagonal solutions in certain diagonal Diophantine systems. Quarterly J. Math. Oxford (2), to appear.Google Scholar
21.F, H. P.. Swinnerton-Dyer. A solution of A5 + B5 + C5 = D5 + E5 + F5. Proc. Cambridge Phil. Soc., 48 (1952), 516–518.Google Scholar
22.Vaughan, R. C. and Wooley, T. D.. On a certain nonary cubic form and related equations. Duke Math. J., 80 (1995), 669–735.CrossRefGoogle Scholar
23.Waerden, B. L. van der. Moderne Algebra, Vol. II, 2nd Ed. (Springer, Berlin, 1940).Google Scholar