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Posts: 234
Registered: 12/4/04
Posted: Feb 22, 2013 8:15 AM
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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
and in addition
a^2 + b^2 + c^2 = 2s.
In this case, the triangle with sides a^2, b^2, and c^2, if constructible (if a^2 + b^2 > c^2, etc.), would be superultraheronic. All our examples generated to satisfy the first equality so far have failed to meet the last one.
If anybody has ideas how they might be generated (say using Mordell theory or some inspired paramterization) or an example, please let me know. A proof the family is empty would also be accepted, but not the welcomed result.
Thanks for your time and consideration!

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