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On the representation of numbers in the formax 2+by 2+cz 2+dt 2

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Acta Mathematica

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Literatur

  1. For the litterature on this subject I refer to the article ofBohr-Cramér (Die neuere Entwicklung der analytischen Zahlentheorie) in the ‘Enzyklopaedie der Mathematischen Wissenschaften’.

  2. ‘Over het splitsen van geheele positieve getallen in een som van kwadraten’, Groningen, 1924.

  3. L. E. Dickson, ‘History of the theory of numbers’, Vol. III (1923), Ch. X.

  4. In my paper ‘On the representation of numbers in the formax 2+by 2+cz 2+dt 2,Proc. London Math. Soc., 25 (1926), 143–173, I have proved some ofLiouville's formulae and some new formulae by means of methods due toHardy andMordell.

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  5. Proc. Camb. Phil. Soc., 19 (1917), 11–21.

  6. A new solution of Waring's problem,Quarterly J. of pure and applied math., vol. 48 (1919), p. 272–293.

  7. Two representationsn=ax 21 +by 21 +cz 21 +dt 21 andn=ax 22 +by 22 +cz 22 +dt 22 will be considered as the same if and only ifx 1=x 2,y 1=y 2,z 1=z 2,t 1=t 2.

  8. ‘On certain trigonometrical sums and their applications in the theory of numbers’,Trans. Camb. Phil. Soc. 22 (1918), 259–276. The formula (1. 56) has already been given byJ. C. Kluyver, ‘Eenige formules aangaande de getallen kleiner dann en ondeelbaar metn’,Versl. Kon. Akad. v. Wetensch., Amsterdam, 1906.

  9. Of course thep 1 occurring here and thep 1 of the lemma's 2, 2*, 3* have quite a different meaning.

  10. It can be proved as follows, thatv 1,V 1 exist. Consider the system of numbersv 1 A 21 +V 1 ϖ 2ξ11 , ifv 1 runs through all numbers, less than and prime toϖ ξ11 andV 1 through all numbers, less than and prime toA 1. Then these numbers are all incongruent modq and they are prime toq. Further the system consists ofϕ(ϖ ξ11 )ϕ(A 1)=ϕ(q) numbers. Therefore one of them must be≡v (modq).

  11. We denote by (M) the number which is ≡M (modq) and for which 0≤(M)<q.

  12. See footnote1 on p. 421. Of course thep 1 occurring here and thep 1 of the lemma's 2, 2*, 3* have quite a different meaning.

  13. Ifλ j ≡o (modϖ), thenS would be 0.

  14. Of course it is also possible, thatS(n j ) tends to zero, but not as quickly as 1/n j , ifn j →∞. But the disenssion of the singular series shows, that in this case, we can always find another sequence, for which the condition 3° holds.

  15. S. Ramanujan, On the expression of a number in the formax 2+by 2+cz 2+dt 2, Proc. Camb. Phil. Soc. 19 (1917), footnote on p. 14.

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  16. Comptes Rendus, Paris, 170 (1920), 354.

  17. Meditationes algebraicae, Cambridge, ed. 3, 1782, 349.

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  1. The Hague

    H. D. Kloosterman

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  1. H. D. Kloosterman
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An account of the principal results of this paper has been published in the ‘Verslagen van de Koninklijke Akademie van Wetenschappen’, Amsterdam, 31 Oct. '25.

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Kloosterman, H.D. On the representation of numbers in the formax 2+by 2+cz 2+dt 2 . Acta Math. 49, 407–464 (1927). https://doi.org/10.1007/BF02564120

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