On the Square Roots of Triangular Numbers

We call an Integer n e Z + a balancing number if 1+ 2+--- + (»-l) = (w + l) + (w + 2) +•• • + ( » +>•) (1) for some r e Z +. Here r is called the balancer corresponding to the balancing number n. For example, 6, 35, and 204 are balancing numbers with balancers 2, 14, and 84, respectively. It follows from (1) that, if n is a balancing number with balancer r, then and thus n2^(n + r)(n + r + l) r =- ( 2 f t + l) + V8ft 2 + l 2 It is clear from (2) that w is a balancing number if and only if n 2 is a triangular number (cf [2], p. 3). Also, it follows from (3) that n is a balancing number if and only if 8n 2 +1 is a perfect square. 2. FUNCTIONS GENERATING BALANCING NUMBERS In this section we introduce some functions that generate balancing numbers. For any balancing number x, we consider the following functions: F(x) = 2xV8x 2 + l, (4) G(x) = 3x + V8x 2 + 1, (5) H{x) = \ + 6V8x 2 +l. (6) First, we prove that the above functions always generate balancing numbers. Theorem 2.1: For any balancing number x, F(x), G(x), and H(x) are also balancing numbers. Proof: Since x is a balancing number, 8x 2 +1 is a perfect square, and 8x 2 (8x 2 + l) ^ 4 x 2 ( 8 x 2 + 1)