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I discuss a variety of results involving s(n), the number of representationsof n as a sum of three squares. One of my objectives is to reveal numerousinteresting connections between the properties of this function and certainmodular equations of degree 3 and 5. In particular, I show thats(25n)=(6-(-n|5))s(n)-5s(n/25) follows easily from the well known Ramanujanmodular 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 quadraticforms 2x^2+2y^2+2z^2-yz+zx+xy and x^2+y^2+3z^2+xy, respectively. I propose aninteresting identity for s(p^2n)- p s(n) with p being an odd prime. Thisidentity makes nontrivial use of the ternary quadratic forms with discriminantsp^2, 16p^2.

On representation of an integer as a sum by X^2+Y^2+Z^2 and the modular equations of degree 3 and 5