Applications of radical method in solving indeterminate equations (Groups)

Author(s): Liu Lushi*

Fermat’s last theorem was proposed by the 17th century French mathematician Pierre de Fermat. He asserted that when the integer n>2, there was no positive solution for the equation.

However, Fermat did not write down his proof, while his other conjectures contributed greatly to mathematics. Therefore, it inspired many mathematicians’ interest in this conjecture. Their corresponding work enriched number theory and promoted its development.

In 1995, Wiles proved that the theorem was valid when n>2. However, his process of proof is tediously long. It is said that only a few world-class masters can understand it, which is confusing.

A perfect cuboid, also known as a perfect box, refers to a cuboid whose edge lengths, diagonals of faces, and body diagonals are all integers. The mathematician Euler once speculated that a perfect rectangle might not exist. No one has been able to prove that it does not exist. What’s a Hellen triangle? A Hellen triangle is a triangle whose sides and areas are rational numbers.

For thousands of years, triangles and their geometric properties have been studied intensively and thoroughly. With the understanding of Hellen triangles, people have found Hellen triangles with three integer heights and with three integer angle bisectors. However, Hellen triangles with three integer midlines have yet to be found.

After several years of research, the author discovered that the above three problems had commonalities and could be demonstrated using the same method. The same algebraic structure is the key to solving these three problems, such as:

The equations y= x³ + ax² + bx + c and y = (x+3)³ + a(x+3)³ + b(x+3) + c are algebraically isomorphic. These two equations represent the same curve and are essentially indistinguishable. The above three problems can be solved with this property easily and concisely.

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