The Fermat's last theorem, the exegesis, Andrew Wiles contribution and the quest for a solution.

A photo of Andrew Wiles


The Fermat's last theorem is a 358 years old theorem in number theory that seeks to resolve problems associates with a three sided triangle.
According to Pythagorean theorem, a triangle whose sides were in the ratio $3:4:5$ would have a right angle as one of its angles. If the length of two sides is squared and added, it will equal the length of the square of the third side i.e $3^2+4^2=9+16=25$ and $5^2=25$. In 1637, Pierre de Fermat stated his conjecture which was a more general equation of the Pythagorean theorem, it states:
"No three positive integer $a,b$ and $c$" satisfy the equation $a^n+b^n=c^n$ for any integer $n>2$". There exists a solution for the cases $n=1$ and $n=2$ which was solved hundreds of years back". 
Using equivalent relations, the Fermat's last theorem can be defined as:
Let $\mathbb{N}$ be the set of Natural numbers and $\mathbb{Z}$ be the set of integers, let $\mathbb{Q}$ be the set of rational numbers $a\b$, where $a$ and $b$ are in $\mathbb{Z}$ with $b\neq{0}$. Then the equation $x^n+y^n=z^n$ where $x,y,z=0$ is a trivial solution and if $x,y,z\neq{0}$ then the solution becomes non-trivial. 

Fermat claimed to have a general proof of his conjecture, but unfortunately left no details of it; not until 30 years after his demise. This claim became known as the Fermat's Last Theorem and has become one of the notable theorems in the history of mathematics and has been listed in the Guinness Book of World Records as the "most difficult mathematics problem" of all time and has remained unsolved for centuries past. 
Fermat's claim proved the conjecture for the primes $3,5,$ and $7$ but the special case of exponent $n=4$ is sufficient to prove the theorem is false for all exponent $n$ that is not prime, therefore, it will also be false for some smaller $n$. 
Does that mean only prime values of $n$ has a solution? 
Let's take a look at the proof of other mathematicians. 
 Sophie Germain proposed her proof for the entire class of primes; while Ernst Kummer proved the theorem for all regular primes and analysing irregular primes individually.
In 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama related elliptic curves and modular forms, this was then known as Taniyama-Shimura conjecture but was later renamed to modularity theorem because of the relationships between the elliptic curves and modular forms that was constructed. To draw the relationship between modularity theorem and Fermat's last theorem, Ken Ribet, Pierre Serre and Frey suggested in their paper that if the modularity conjecture could be proven for semi-stable class of elliptic curves, then a proof of Fermat's last theorem would follow immediately. 
Diophantine's equation  studied a problem where two integers $x$ and $y$ such that, their sums and the sum of their squares equals two given numbers $A$ and $B$ respectively:
$A=x+y$ and $B=x^2+y^2$, this approach to Fermat's last theorem seeks to proof the equation with positive integer solution. Only one proof by Fermat has survived over the course of this travail which is the case for $n=4$ where the technique of infinite descent is used to show the area of a right angle triangle with integer sides can never equal the square of an integer. This is formulated as thus: $x^4-y^4=z^2$, and it has no primitive solution in integers and this shows that the equation $a^4+b^4=c^4$ can be rewritten as $c^4-b^4=(a^2)^2$, alternative methods were later developed by many mathematicians. 
Prizes have been attached to this proof since the 18th century. The French Academy of sciences offered a price for the general proof of Fermat's last theorem and 3000 francs and a gold medal was awarded to Kummer for his research on ideal numbers. 
The academy of Brussels offered another price in 1883.
German mathematician Paul Wolfskehl offered 100,000 gold marks through the Gottingen Academy of sciences in 1908 to anyone who could complete proof of Fermat's last theorem, the price was governed by nine rules which was published in national dailies for awarding the price. 

A photo of Andrew Wiles

Andrew Wiles later won the Wolfskehl price in 2007 which was now worth $50,000. In addition, he was also awarded an Abel price by the Norwegian government for his stunning proof of Fermat's last theorem which was worth €600,000. He used the modularity conjecture for semi-stable elliptic curves, this was the beginning of a new era in number theory. Prior to Wiles proof, thousands of incorrect proofs have been submitted to the wolfskehl committee. 

Photo of The Simpson's TV series


The Fermat's last theorem has appeared in notable TV series such as "The Simpson" in the episode "The wizard of evergreen terrace" and   has also appeared on "Star Trek: The Next Generation".