Why is every positive integer the sum of 3 triangular numbers?
By Michael Henderson •
Why is every positive integer the sum of at most 3 triangular numbers ?
$\endgroup$ 51 Answer
$\begingroup$Every positive integer $\equiv 3\mod{8}$ can be written as a sum of three squares; see here for a proof (in fact, more integers than just those can be so written).
The result about triangular numbers follows from that result: let $n>0$; then $8n+3$ is a sum of three squares. From congruence conditions modulo $4$, it follows that each square is odd, so that $$ 8n+3 = (2x+1)^2 + (2y+1)^2 + (2z+1)^2 = 4x^2 + 4x + 4y^2 + 4y + 4z^2 + 4z + 3,$$ so that $$ 8n = 4x(x+1) + 4y(y+1) + 4z(z+1).$$ The result follows upon dividing through by $8$.
$\endgroup$ 5
