ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11, 51 Khan, Nazir, GIK Institute of Engg. Sc. & Tech., Topi, Pakistan The Numbers that Can Be Represented as the Sum of Four Squares (One Repeating Itself Three Times), in Three Different Ways All numbers of the form 4(n^{2} + 3n + 3) (n = 1, 2, 3, ..) can be represented in three different ways, viz:

3.1^{2} + (2n + 3)^{2} = 3(n + 1)^{2} + (n + 3)^{2} = 3(n + 2)^{2} + n^{2}

The fact that the first square repeats itself three times leads to its geometrical representation in three dimensions. It gives a set of circles, with centre at the point (1, 1, 1), cut by the plane x + y + z = 3 on the sphere x^{2} + y^{2} + z^{2} = 28, [which is 4(n^{2} + 3n + 3), for n = 1]. It will also represent a family of pairs of circles lying in the planes parallel to the aforesaid plane, at intervals of 3^{1/2}, with their centres lying on the line x = y = z, which will be cut on the successive spheres x^{2} +y^{2} +z^{2} = 4(n^{2} +3n+3). A sub-class of these numbers can also be represented, retaining the same characteristics, in six different ways.