## Acta Mathematica

- Acta Math.
- Volume 49, Number 3-4 (1926), 407-464.

### On the representation of numbers in the form *ax*^{2}+*by*^{2}+*cz*^{2}+*dt*^{2}

**Full-text: Open access**

#### Note

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.

#### Article information

**Source**

Acta Math., Volume 49, Number 3-4 (1926), 407-464.

**Dates**

First available in Project Euclid: 14 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.acta/1487102066

**Digital Object Identifier**

doi:10.1007/BF02564120

**Rights**

1926 © Almqvist & Wiksells Boktryckeri-A.-B.

#### Citation

Kloosterman, H. D. On the representation of numbers in the form ax 2 + by 2 + cz 2 + dt 2. Acta Math. 49 (1926), no. 3-4, 407--464. doi:10.1007/BF02564120. https://projecteuclid.org/euclid.acta/1487102066

#### Literatur

- For the litterature on this subject I refer to the article of Bohr-Cramér (Die neuere Entwicklung der analytischen Zahlentheorie) in the ‘
*Enzyklopaedie der Mathematischen Wissenschaften*’. - ‘Over het splitsen van geheele positieve getallen in een som van kwadraten’, Groningen, 1924.
- L. E. Dickson, ‘History of the theory of numbers’, Vol. III (1923), Ch. X.
- In my paper ‘On the representation of numbers in the form
*ax*^{2}+*by*^{2}+*cz*^{2}+*dt*^{2},*Proc. London Math. Soc.*, 25 (1926), 143–173, I have proved some of Liouville's formulae and some new formulae by means of methods due to Hardy and Mordell. *Proc. Camb. Phil. Soc.*, 19 (1917), 11–21.- A new solution of Waring's problem,
*Quarterly J. of pure and applied math.*, vol. 48 (1919), p. 272–293. - Two representations
*n=ax*${}_{1}^{2}$ +*by*${}_{1}^{2}$ +*cz*${}_{1}^{2}$ +*dt*${}_{1}^{2}$ and*n=ax*${}_{2}^{2}$ +*by*${}_{2}^{2}$ +*cz*${}_{2}^{2}$ +*dt*${}_{2}^{2}$ will be considered as the same if and only if*x*_{1}=*x*_{2},*y*_{1}=*y*_{2},*z*_{1}=*z*_{2},*t*_{1}=*t*_{2}. - ‘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 by J. C. Kluyver, ‘Eenige formules aangaande de getallen kleiner dan*n*en ondeelbaar met*n*’,*Versl. Kon. Akad. v. Wetensch., Amsterdam*, 1906. - Of course the
*p*_{1}occurring here and the*p*_{1}of the lemma's 2, 2^{*}, 3^{*}have quite a different meaning. - It can be proved as follows, that
*v*_{1},*V*_{1}exist. Consider the system of numbers*v*_{1}*A*${}_{1}^{2}$ +*V*_{1}*ϖ*${}_{1}^{2ξ1}$ , if*v*_{1}runs through all numbers, less than and prime to*ϖ*${}_{1}^{ξ1}$ and*V*_{1}through all numbers, less than and prime to*A*_{1}. Then these numbers are all incongruent mod*q*and they are prime to*q*. Further the system consists of*ϕ*(*ϖ*${}_{1}^{ξ1}$ )*ϕ*(*A*_{1})=*ϕ(q)*numbers. Therefore one of them must be≡*v*(mod*q*). - We denote by (
*M*) the number which is ≡*M*(mod*q*) and for which 0≤(*M*)<*q*. - See footnote
^{1}on p. 421. Of course the*p*_{1}occurring here and the*p*_{1}of the lemma's 2, 2^{*}, 3^{*}have quite a different meaning. - If
*λ*_{j}≡o (mod*ϖ*), then*S*would be 0. - Of course it is also possible, that
*S(n*_{j}) tends to zero, but not as quickly as 1/*n*_{j}, if*n*_{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. - S. Ramanujan, On the expression of a number in the form
*ax*^{2}+*by*^{2}+*cz*^{2}+*dt*^{2}, Proc. Camb. Phil. Soc. 19 (1917), footnote on p. 14. - Comptes Rendus, Paris, 170 (1920), 354.
- Meditationes algebraicae, Cambridge, ed. 3, 1782, 349.

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