Dear Gerald:

The chapter 14, The Eternal Triangle, in Recreations in the Theory of Numbers by A.H.Beiler, is wholly devoted to Pythagorean triples with various additional properties. Specifically, on p 125, there appears a discussion on the triples with consecutive legs.

Assume c_r is the hypotenuse of the r-th triple (ordered by, say, the smallest leg, or by hypotenuse), Then c_r is given by

[(sqrt(2)+1)^(2r+1) + (sqrt(2)-1)^(2r+1)]/(2sqrt(2))

which explains your observation with regard to the factor (sqrt(2)+1)².

To obtain the formula Beiler refers to a recurrence relation that can be surmised from the first few such triples and then proven by induction.

Following is the URL of the amazon.com page with the book's description:

http://www.amazon.com/exec/obidos/ISBN=0486210960/001-7810978-3759339

Best regards,
Alexander Bogomolny

Copyright © 1996-2009 Alexander Bogomolny

31182838