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

Every positive integer can be expressed as a sum of three triangular numbers.

Proof

The triangular numbers are

\[T_r= \frac{r(r+1)}{2}, \; r \ge 1\]

.\[n=T_r+T_s+T_t= \frac{r(r+1)}{2}+ \frac{s(s+1)}{2}+ \frac{t(t+1)}{2}\]

\[8n+3=4r(r+1)+4s(s+1)+4t(t+1)+3==(2r+1)^2+(2s+1)^2+(2t+1)^2\]