Prove that there are an infinite number of distinct ordered pairs (m, n) of integers such that, for every positive integer t, the number mt + n is a triangular number if and only if t is a triangular number as well

If I assume m and n are positive I can't find any numbers that work. 

It appears I am trying to prove there are triangular numbers whose congruence class (mod m) is unique.  I don't think there are any.

 

If we allow m to be negative, then the conditions hold if n=any triangular number and m=-n.

This seems too easy.  (It may also beg the question of whether 0 or negatives can be triangular.  The statement that t is positive really makes me wonder.)

-Jer

Posted by Jer on 2004-10-26 17:33:44