## Every Positive Integer Van Be Written as the Sum of Three Triangular Numbers

Theorem
Every positive integer can be expressed as a sum of three triangular numbers.
Proof
The triangular numbers are
$T_r= \frac{r(r+1)}{2}, \; r \ge 1$
.
$n=T_r+T_s+T_t= \frac{r(r+1)}{2}+ \frac{s(s+1)}{2}+ \frac{t(t+1)}{2}$

$8n+3=4r(r+1)+4s(s+1)+4t(t+1)+3==(2r+1)^2+(2s+1)^2+(2t+1)^2$

Hence
$8n+3$
can be written as the sum of three squares. Reversing the process expresses
$n$
as the sum of three triangular numbers.
Integers cannot be simultaneously of the form
$8n_3$
and
$4^n(8k+7)$
(No Number of Form 4^n(8m+7) Can Be Written as a Sum of Three Squares) so the theorem is proved.