Perfect Triangles: Rational Points on the Curve C4 (The Unsolved Case)

Abstract

Abstract Finding a perfect triangle was stated as an open problem by Guy in [6]. Numerous researches have been done in the past to find such a triangle, unfortunately, to date, no one has ever found one, nor has proved its non-existence. However, on the bright side, there are partial results which show that there exist triangles that satisfy five or even six of the seven parameters to be rational. In this paper, we perform an extensive search to investigate if we can extract any perfect triangles from the curve C4 based on the final unsolved case in [9], which will then complete the proof of existence or non-existence of perfect triangle on the curve. Multiple conjectures were tested to eliminate the possibilities of finding a perfect triangle from the last unsolved case of n ≡ 3024 (mod 6052) in [9]. Finally, a theorem was proved, which was subtle enough to eliminate this case, proving that there does not exist any perfect triangle arising from the curve C4.