On Representation of an Integer by X ² + Y ² + Z ² and the Modular Equations of Degree 3 and 5

Abstract

I discuss a variety of results involving s(n), the number of representations of n as a sum of three squares. One of my objectives is to reveal numerous interesting connections between the properties of this function and certain modular equations of degree 3 and 5. In particular, I show that

$$\displaystyle{ s(25n) = \left (6 -\left (-n\vert 5\right)\right)s(n) - 5s\left (\frac{n} {25}\right) }$$

follows easily from the well known Ramanujan modular equation of degree 5. Moreover, I establish new relations between s(n) and h(n), g(n), the number of representations of n by the ternary quadratic forms

$$\displaystyle{2{x}^{2} + 2{y}^{2} + 2{z}^{2} - yz + zx + xy,\quad {x}^{2} + {y}^{2} + 3{z}^{2} + xy,}$$

respectively. Finally, I propose a remarkable new identity for s(p ² n)−p s(n) with p being an odd prime. This identity makes nontrivial use of the ternary quadratic forms with discriminants p ², 16p ².

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