Express the following as sum of four squares:
(a) $3072 = 2^{10} \times 3$
(b) $4992 = 2^7 × 3 × 13$
(c) $2015$
(d) $2016$
(e) $2020$
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)
$$ 3072 = (32)^2 \times 3 = (32)^2 \times (1^2 + 1^2 +1^2 + 0^2) = 32^2 + 32^2 + 32^2 + 0^2$$
(b)
$$ \begin{align} 4992 & = 2^6 \times 6 \times 13 = 2^6 \times (2^2 + 1^2 + 1^2 + 0^2) \times (3^2 + 2^2 + 0^2 + 0^2) \\ \\ & = 8^2 \times (8^2 + 3^2 + 1^2 + 2^2) = 64^2 + 24^2 + 8^2 + 16^2 \end{align} $$
(c)
$$ 2015 = 65 \times 31 = (8^2 + 1^2 + 0^2 + 0^2) \times (5^2 + 2^2 + 1^2 +1^2) = 42^2 + 7^2 + 11^2 + 9^2 $$
(d)
$$ 2016 = 32 × 63 = (4^2 + 4^2 + 0^2 + 0^2) \times (7^2 + 3^2 +2^2 + 1^2) = 40^2 + 4^2 + 16^2 + 12^2 $$
(e)
$$ 2020 = 20 × 101 = (4^2 + 2^2 + 0^2 +0^2) \times (10^2 + 1^2 + 0^2 + 0^2) = 42^2 + 16^2 + 0^2 + 0^2 $$