Convert the following integers into sum of four squares:
(a) 217
(b) 819
(c) 2109= 3 × 19 × 37
We'll be using the Sum of Four Squares Identity (8.10) is
$$ \begin{align} (a^2 + b^2 + c^2 + d^2) \times (x^2 + y^2 + z^2 + w^2) & = (ax + by + cz + dw)^2 + (az− bw− cx + dy)^2 \\ \\ & + (ay− bx + cw− dz)^2 + (aw + bz− cy− dx)^2 \end{align} $$
(a)
$$ 217= 7 \times 31 = (2^2 +1^2 + 1^2 + 1^2) \times (5^2 + 2^2 + 1^2 + 1^2) = 14^2 + 2^2 + 1^2 + 4^2 $$
(b)
$$ 819 = 3^2 × 7 × 13 = (2^2 + 1^2 + 1^2 + 1^2) \times (9^2 + 6^2 + 0^2 + 0^2) = 24^2 + 3^2 + 3^2 + 15^2 $$
(c)
$$ 2109 = 57 \times 37 = (7^2 + 2^2 + 2^2 + 0^2) \times (6^2 + 1^2 + 0^2 +0^2) = 44^2 + 12^2 + 5^2 + 2^2 $$