## Any integer n; n = i^2 + j^2 – 5k^2

For any integer   $n$   there exists integers   $i, \; j$,   and   $k$   such that

$n \; = \; i^2 \; + \; j^2 \; - \; 5 \,k^2$

$(m - a_1)^2 + (2 \, m - b_1)^2 - 5(m - c_1)^2 \; = \; 2 \, m \; = \; n$
$(m - a_2)^2 + (2 \, m - b_2)^2 - 5(m - c_2)^2 \; = \; 2 \, m \; + \; 1 \; = \; n$

$a_1 \; = \; 2 \,(5 \, k^2 + 5 \, k + 1)$
$b_1 \; = \; 20 \, k^2 + 10 \,(2 - t) \,k + (6 - 5 \, t)$
$c_1 \; = \; 10 \, k^2 + 2 \,(5 - 2 \, t) \,k + (3 - 2 \, t)$

$a_2 \; = \; 10 \, k^2 \; - \; 1$
$b_2 \; = \; 10 \,k \,(2 \,k + t)$
$c_2 \; = \; 2 \,k \,(5 \,k + 2 \,t)$

with   $t = \pm 1$   and   $k$   any integer.