Convert the following integers into sum of four squares by using the Sum of Four Squares Identity (8.10).
(a) 35 (b) 49 (c) 945 (d) 310 (e) 465 (f) 143
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)
$$ \begin{align} 35 = 5 \times 7 & = (2^2 + 1^2 + 0^2 + 0^2) \times (2^2 + 1^2 +1^2 +1^2) \\ \\ & = (4 + 1)^2 + (2 - 1)^2 +(2-2)^2 +(2+1)^2 \\ \\ & = 5^2 + 1^2 + 0^2 + 3^2 \end{align}$$
(b)
$$ 49 = 7 \times 7 = (2^2 + 1^2 +1^2 +1^2) \times (2^2 + 1^2 +1^2 +1^2) = 7^2 + 0^2 + 0^2 + 0^2 $$
(c)
$$ 945 = 9 \times 105 = (3^2 + 0^2 +0^2 +0^2) \times (10^2 + 2^2 +1^2 +0^2) = 30^2 + 3^2 + 6^2 + 0 $$
(d)
$$ 310 = 10 \times 31 = (3^2 + 1^2 + 0^2 +0^2) \times (5^2 + 2^2 + 1^2 + 1^2) = 17^2 + 2^2 + 1^2 + 4^2 $$
(e)
$$ 465 = 15 \times 31 = (3^2 + 2^2 +1^2 + 1^2) \times (5^2 + 2^2 + 1^2 + 1^2) = 21^2 + 2^2 + 4^2 + 2^2 $$
(f)
$$ 143 = 11 \times 13 = (3^2 + 1^2 + 1^2 + 0) \times (3^2 + 2^2 + 0^2 +0^2) = 11^2 + 3^2 + 3^2 + 2^2 $$