A harmonic sequence is defined as a sequence whose reciprocals form an arithmetic sequence. For example: ({10, 12, 15, 20} is harmonic since {1/10, 1/12, 1/15, 1/20} is arithmetic. (Reference:http://perplexus.info/show.php?pid=1557)

Determine three positive integers X,Y and Z satisfying X < Y < Z which belong to a Harmonic sequence with each of X, Y and Z being expressible as the sum of squares of two distinct positive integers such that X + Y = Z + 1 , with Z being less than 2006.

From

(1) 1/y - 1/z = 1/x - 1/y

(2) x + y = z +1

we have

(3) y = (1/2) (1 + 4z - sqrt(1 + 8 z^2))

For z < 2006 and integer solutions of the square root we have possible values for z of {1, 6, 35, 204, 1189} of which only 1189 is the sum of two squares.

So we have the solution {x, y, z} = {493, 697, 1189}.

Posted by goFish on 2006-01-31 17:15:12