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
, we'll prove that m=9, n=1 satisfy the given conditions.
If t is triangular, t=n(n+1)/2, and mt+n=9n(n+1)/2+1 = (3n+1)(3n+2)/2, which is triangular. Vice versa, if 9t+1 is triangular, then 9t+1=n(n+1)/2, so n=3k+1 [n=3k or n=3k-1 don't produce a 9t+1 result] and we have 9t+1=9k(k+1)/2+1, so t is triangular.
Second, if m=p and n=q are a solution, so are m=pē and n=(p+1)q, so we get an infinite number of such pairs.
Edited on October 26, 2004, 6:25 pm