The triangular numbers are 1, 3, 6, 10, ... , n(n+1)/2, ... . Prove that there are an infinite number of pairs ai, bi such that m is triangular iff aim + bi is triangular.
Let Tn = n(n + 1)/2. Let us try to express Tan+b in the form ATn + B. Evidently this works if a = 2b + 1, with A = (2b + 1)2 and B = b(b + 1)/2.
The only if part is not quite so obvious. Suppose that T = (2b + 1)2t + b(b+1)/2 is triangular. We must show that t is triangular. Put k = b(b+1). Then T = (4k+1)t + k/2. Since T is triangular, 8T+1 = 8(4k+1)t + (4k+1) is an odd square. But 4k+1 = (2b+1)2 is an odd square, so 8t+1 is an odd square and hence t is triangular also.
49th Putnam 1988
© John Scholes
1 Jan 2001