## Pierre one succeeded. It was known as Fermat’s

Pierre de Fermat was a French lawyer during the 1600s. He made numerous discoveries in different mathematical subjects: analytical geometry, probability, differential calculus, number theory, and optics. In Diophantus’ Arithmetica, there is one of his works that many high-leveled mathematicians tried to solve for years, but no one succeeded. It was known as Fermat’s Last Theorem. In short, Fermat stated that there are no natural numbers x, y, z that correspond to the equation xn + yn = zn with any value n that is greater than 2. In other words, some squared numbers are the sum of two other squared numbers. But this case never happens for any other number that is in the power of more than two. It seemed almost impossible to solve the theorem.
But in 1995, Andrew Wiled proved the theorem with the help of other mathematicians. His proof is exceptionally long and difficult.
Even before Fermat was born, many important mathematical theories had been invented, which later helped in the solving of Fermat’s Last Theorem. The first important invention was Poincaré’s modular forms. His work resembled some of Galois’s theory and Euler’s works. His modular forms are periodic, which means it repeats itself. The modular forms are great ways to convert infinite variations to finite. That means it can convert variations that are almost uncountable to a set of countable variations. The second great invention was done by Gauss. Gauss invented the elliptic curve. Elliptic curves are curved lines on a graph that contains a definite slope. The third great invention was done by Yutaka Taniyama and Goro Shimura of Japan. Their invention later became known as the Taniyama-Shimura Theorem. They concluded that elliptical forms are endless up to infinite. But the elliptic curve over the rational numbers can be uniformed by a modular form. As modular forms are specific, their infinite can be converted to finite and thus the uncountable can become countable. Their conclusion was first not accepted. But Gerhard Frey researched and found that if Taniyama and Shimura were wrong, then there will be an existing curve that looks very random and impossible to exist. His curve was known as Frey’s curve. After peer review, Frey concluded that Frey’s curve is impossible to exist. And if Frey’s curve does not exist, then Taniyama-Shimura Theorem must lead to the solving of Fermat’s Last Theorem.
Even though Taniyama-Shimura Theorem is said to be leading towards the solving of Fermat’s Last Theorem, it was impossible for mathematicians to conclude how Taniyama-Shimura Theorem leads to the solving of Fermat’s Last Theorem. In the 1990s, an American mathematician, Andrew Wiles, expressed his desire to solve Fermat’s Last Theorem. In his childhood, Andrew found the book about the theorem at the school library, and fascinated by the fact he could actually understand it, set out to be the first to prove it. During the 1990s, Andrew gave his time towards viewing different theorems and conjectures that will help him in solving Fermat’s Last Theorem. When he was searching, he found the Taniyama-Shimura Conjecture, which was appealing to him. Wiles knew that every elliptic curve is modular, but he has to prove it. For example, he has to show that every elliptic curve is really a modular form in disguise. But nobody had any idea how to show such a strange connection between these two seemingly very different entities.
Later, Wiles concluded that some of the elliptic curves are already said to be modular. But the problem is there are infinitely many modular forms and elliptic curves. The solution was in his head; Wiles used Galois Theory to transform the infinity to a finite form. Now he can compare the elliptic curves and the modular forms in a finite way. During his work, Wiles faced a major problem. He cannot find all the modular forms for the elliptic curves that depend upon prime numbers. So, he took help of another great mathematician, Berry Mazur. They met in a café and Wiles proposed him for help in the comparison of modular forms for the elliptic curves that depend upon prime numbers. In answer, Mazur said that if he can find one modular form for the elliptic curve that depends upon a prime number, it has to be true for another prime number. For example, the three-to-five switch; elliptic curves based on prime number three can be transformed to prime number five. This was what Wiles needed.
Wiles finished his work on Fermat’s Last Theorem and sent it for peer review. But a subtle error was discovered. In the way he showed, the Euler System does not make any contribution towards the theorem. It made the total theorem unsolved. Therefore, Wiles abandoned from doing any further work another year to figure it out. In 1995, when viewing he found that Euler System is a failure, but it makes the Horizontal Iwasawa Theory which supports the Fermat’s Last Theorem. This was a golden moment for Wiles. Now he sent his explanation for peer review At this time, his solution was perfectly accepted without any mistakes. Andrew Wiles solved the mystery of Fermat’s Last Theorem. After more than three-hundred years the Fermat’s Last Theorem has been solved. But nobody knows until this day whether Fermat knew the proof of his theorem and if there is another method of solving it.