Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-10T04:27:13.023Z Has data issue: false hasContentIssue false
  • English
  • Français

ON WARING’S PROBLEM: TWO SQUARES AND THREE BIQUADRATES

Part of: Additive number theory; partitions Multiplicative number theory

Published online by Cambridge University Press:  28 June 2013

John B. Friedlander  and
Trevor D. Wooley
John B. Friedlander
Affiliation:
Department of Mathematics, University of Toronto, Toronto ON,Canada, M5S 2E4 email frdlndr@math.toronto.edu
Trevor D. Wooley
Affiliation:
School of Mathematics, University of Bristol, University Walk, Clifton, Bristol BS8 1TW,U.K. email matdw@bristol.ac.uk

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott–Halberstam conjecture, we show that all large natural numbers $n$ with $8\nmid n$, $n\not\equiv 2~(\text{mod} ~3)$ and $n\not\equiv 14~(\text{mod} ~16)$ are the sum of two squares and three biquadrates.

MSC classification

Secondary: 11P05: Waring's problem and variants 11N36: Applications of sieve methods 11P55: Applications of the Hardy-Littlewood method

Information

Type
Research Article
Information
Mathematika , Volume 60 , Issue 1 , January 2014 , pp. 153 - 165
Copyright
Copyright © University College London 2013 

References

Brüdern, J., Sums of squares and higher powers. J. Lond. Math. Soc. (2) 35 (1987), 233–243.CrossRefGoogle Scholar
Brüdern, J., A problem in additive number theory. Math. Proc. Cambridge Philos. Soc. 103 (1988), 27–33.CrossRefGoogle Scholar
Davenport, H., Analytic Methods for Diophantine Equations and Diophantine Inequalities, 2nd edn. Cambridge University Press (Cambridge, 2005).Google Scholar
Dietmann, R. and Elsholtz, C., Sums of two squares and one biquadrate. Funct. Approx. Comment. Math. 38 (2008), 233–234.CrossRefGoogle Scholar
Duke, W., Friedlander, J. B. and Iwaniec, H., Weyl sums for quadratic roots. Int. Math. Res. Not. 2012 (11) (2012), 2493–2549; Corrigendum, Int. Math. Res. Not. 2012(11) (2012), 2646–2648.Google Scholar
Friedlander, J. B. and Iwaniec, H., Opera de Cribro, American Mathematical Society (Providence, RI, 2010).Google Scholar
Gauss, C. F., Disquisitiones Arithmeticae (Leipzig, 1801).CrossRefGoogle Scholar
Golubeva, E. P., On nonhomogeneous Waring equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 226 (1996), Anal. Teor. Chisel i Teor. Funktsii. 13, 65–68, 236; translation in J. Math. Sci. (New York) 89 (1998), 955–957.Google Scholar
Golubeva, E. P., A bound for the representability of large numbers by ternary forms, and nonhomogeneous Waring equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 357 (2008), Anal. Teor. Chisel i Teor. Funktsii. 23, 5–21, 224; translation in J. Math. Sci. (New York) 157 (2009), 543–552.Google Scholar
Hooley, C., On a new approach to various problems of Waring’s type. In Recent Progress in Analytic Number Theory, Vol. 1 (Durham, 1979), Academic Press (London–New York, 1981), 127–191.Google Scholar
Hooley, C., On Waring’s problem for two squares and three cubes. J. Reine Angew. Math. 328 (1981), 161–207.Google Scholar
Hooley, C., On Waring’s problem for three squares and an lth power. Asian J. Math. 4 (2000), 885–903.CrossRefGoogle Scholar
Kawada, K. and Wooley, T. D., Sums of fourth powers and related topics. J. Reine Angew. Math. 512 (1999), 173–223.CrossRefGoogle Scholar
Kawada, K. and Wooley, T. D., Davenport’s method and slim exceptional sets: the asymptotic formulae in Waring’s problem. Mathematika 56 (2010), 305–321.CrossRefGoogle Scholar
Linnik, Yu. V., Additive problems involving squares, cubes and almost primes. Acta Arith. 21 (1972), 413–422.CrossRefGoogle Scholar
Vaughan, R. C., On sums of mixed powers. J. Lond. Math. Soc. (2) 3 (1971), 677–688.Google Scholar
Vaughan, R. C., On Waring’s problem for smaller exponents. Proc. Lond. Math. Soc. (3) 52 (1986), 445–463.Google Scholar
Vaughan, R. C., The Hardy–Littlewood Method, 2nd edn. Cambridge University Press (Cambridge, 1997).CrossRefGoogle Scholar