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.
