From: jpr2718@aol.com (Jpr2718)
Subject: Re: Triangle Areas in Progression
Date: 27 May 1999 03:35:27 GMT
Newsgroups: sci.math
Keywords: Another elliptic surface 

sestnick@ariel.com (Steve Estnick) wrote:

> Can the areas of three rational right
>  triangles on the same hypotenuse
>  be in arithmetic progression?

This is equivalent to the question as to whether there can be two 3-term
arithmetic progressions with the same differnce between terms, so that every
entry in both progressions is an integer square, and the average of the two
central terms is also a square.

This is an open problem.

Kevin Brown has shown that if there is such a solution, then the square that is
the average of the two central terms can be written as a sum of squares in more
than 4 ways.  See the entry on magic squares of squares at 
http://www.seanet.com/~ksbrown/inumber.htm

John