Fermat's Last Theorem: An Elegant Problem In Mathematics

Today, it's 418th birth anniversary of the man who blessed the world with a problem that took 358 years to be solved. Often called the founder of the modern theory of numbers, "Pierre de Fermat" was born in Beaumont-de-Lomagne, France. Although, he was a lawyer by profession and just an amateur mathematician. Still, Fermat was one of the two leading mathematicians of the first half of the 17th century. Here, we will try to learn about the famous Fermat's last theorem, that troubled the mathematical community for 358 years.

Fermat's Last Theorem
Pierre De Fermat
Image Courtesy: Wikipedia.org

What is Fermat's last theorem?

Fermat's last theorem states that no three positive integers p, q, and r satisfy the equation:

pª + qª = rª

for any integer value of a greater than 2. The cases a = 1 and a = 2 have been known since ancient times to have an infinite number of solutions.

Story behind Fermat's last theorem :

Fermat always refused to publish his work. So, his friends and loved ones always feared that his work would soon be forgotten unless something was done about it. In order to do something regarding this issue, his son, Samuel, did something. He started collecting Fermat's letters and other mathematical papers. He also found his comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way only, the famous 'Last theorem' came to be published. Samuel found it written as a marginal note in his father's copy of Diophantus's Arithmetica. This also included a note by Fermat which says:
"I have discovered a truly remarkable proof which this margin is too small to contain". Hence, only the theorem was found, but the proof for the theorem by Fermat was never found.

Also Read: Top 10 Quotes on Quantum Mechanics

Attempts to find the proof :

Many attempts were made time and again to prove Fermat's last theorem. But, the things were not as simple as they seem to be. Mathematicians toiled hard for more than 3 centuries to have a satisfying proof of this famous theorem. No doubt, there were first many speculations about the authenticity of this theorem as the publication was done by Fermat's son without his consent, after his death. But, after 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles. This proof was formally published in 1995. Even prior to Wiles's proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet (3 meters) of correspondence.

Fermat's Last Theorem
Andrew Wiles
Image Courtesy: npr.org

Wiles' proof of the Fermat's last theorem:

In his much awaited proof, Wiles has used many techniques from algebraic geometry and number theory. It has many outcomes in these branches of mathematics. This proof is also based on the standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques which were not available to Fermat. Wiles' proof was compiled in two papers. Together, these papers were 129 pages long. This consumed over seven years of Wiles' research time. John Coates described the proof as one of the highest achievements of number theory, and John Conway called it the proof of the century.

Prior to its proof, this theorem made headlines in the Guinness Book of World Records for being the "most difficult mathematical problem" as it had the largest number of unsuccessful proofs. Many prizes have been offered time and again in order to attract people to search for a proof for this theorem . Undoubtedly, Fermat's last theorem is among the most notorious and notable theorems in the history of mathematics.

Also read: How to end your struggle with mathematics and finally get it right.

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