Copyright ©2008 by Paul Niquette. All rights reserved.. 
he
puzzle
asks for the formula that generates a family of right
triangles having
the indicated Natural Hypoteneese
starting
with 345, 51213, 72425, 94041. Two
observations:
You might enjoy proving that these formulas work for all values of n... [2n +1]^{2} + [2n(n + 1)]^{2} = [2n(n + 1) +1]^{2} You might also enjoy exploring the question of how many Natural Hypoteneese there are. Well, of course, there is an infinite number  in each family. Still, there are plenty of integers that are not Natural Hypoteneese. The graph below shows the cumulative Natural Hypoteneese in relation to the natural numbers. You are invited to send your favorite
observations about
Natural
Hypoteneese here.




