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.