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Topic: DO SUPERULTRAHERONIC TRIANGLES EXIST?
Replies: 0

 wheierman@corunduminium.com Posts: 234 Registered: 12/4/04
DO SUPERULTRAHERONIC TRIANGLES EXIST?
Posted: Feb 22, 2013 8:15 AM

A heronic triangle is ordinarily defined as one one with integer side-lengths and integer area. The family includes the pythagorean triangles (right triangles with integer side-lengths).
Let us define an ultraheronic triangle as a heronic triangle for which each factor (s, s-a, s-b, and s-c) in the radicand of Heron's formula is a square. These are not too hard to find. One example is the "25-153-160" triangle (s=169, s-a=144, s-b=16, s-c=9).
Let a superultraheronic triangle be an ultraheronic triangle for which the sidelengths themselves are also squares. We note in the above example that one side is a square, but that is the best we have been able to do so far.
We have tried the following number-theoretic approach. Suppose the semiperimeter is a number that can be written as a sum of two squares in three ways (at least two of which must be different). This would be the case if for example the semiperimeter s is the square of a product of two or more Fermat primes (primes congruent to 1 (mod 4)). We would like to have examples where
s = a^2 + x^2 = b^2 + y^2 = c^2 + z^2