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Let s(n) be the number of representations of n as the sum of three squares.We prove a remarkable new identity for s(p^2n)- ps(n) with p being an oddprime. This identity makes nontrivial use of ternary quadratic forms withdiscriminants p^2 and 16p^2. These forms are related by Watson'stransformations. To prove this identity we employ the Siegel--Weil and theSmith--Minkowski product formulas.

On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2