[Maths Class Notes] on Fermat's Last Theorem Pdf for Exam

Fermat’s last theorem states that no three positive integers, say, x, y, and z will satisfy the equation xⁿ + yⁿ = zⁿ for any integer value of n greater than 2. Since ancient times, the equation for n=1 and n=2 has been well-known to hold infinitely many solutions. Sometimes, this theorem is also known as Fermat’s Conjecture. Pierre de Fermat stated this proposition as a theorem about 1637 and stated that he had proof that did not fit in the margin. Some of the statements claimed by Fermat without proof were later proven by other mathematicians and credited as Fermat’s theorem. However, the last theorem of Fermat resisted proof, leading to doubt that it was ever having a correct proof. Let us acknowledge who gave the proof of Fermat’s conjecture, equation, and other concepts related to the theorem.

Equation of the Last Theorem Stated by Fermat

x²+ y² = z² is a Pythagorean equation that has an infinite number of solutions for different values of x, y, and z. These solutions refer to Pythagorean triples. Fermat’s theorem states that the general equation xⁿ + yⁿ = zⁿ has no solutions for positive integers if n is a natural number greater than 2. For instance, if n=3, then according to the theorem, no such x, y, and z natural number exists for which x³+ y³ = z³. It implies that a cube cannot be a sum of two cube numbers. 

According to the last theorem, there exists no natural number greater than 2 for which the equation xⁿ + yⁿ = zⁿ satisfies. 

However, Fermat left no details of the proof, and his claim was discovered after his death. This claim became Fermat’s enigma, which stood unsolved for some centuries. 

Fermat’s Last Theorem Proof Simplified

The proof of both Modular elliptic curves and Fermat’s last theorem were considered inaccessible to proof by mathematicians. Wiles announced the proof at a lecture entitled Modular Forms, Elliptic Curves, and Galois Representations in 1993. He proved the theorem by contradiction in which he assumes the opposite of what is required to prove. The contradiction shows that the taken assumption was incorrect and the statement was correct. The proof follows two parts in which the first part involves a general result about lifts. It refers to the modularity lifting theorem, and the proof of Fermat’s last theorem can be mathematically written as xⁿ + yⁿ = zⁿ

For n=2, Fermat equation can be stated as: x²+ y² = z². 

A first attempt to get Fermat’s last theorem solution can be made by factoring the equation, that is, (z^n/2 + y^n/2 ) (z^n/2 – y^n/2) = xⁿ.

As the power is an exact power, the equation gives:

z^n/2 + y^n/2 = 2^n-1 pⁿ

z^n/2 – y^n/2 = 2 qⁿ

Now, solving for the values of y and z, the equation becomes:

z^n/2 = 2^n-2 pⁿ + qⁿ

y^n/2 = 2^n-2 pⁿ – qⁿ

It gives:

z = (2^n-2 pⁿ + qⁿ)^2/n

y = (2^n-2 pⁿ – qⁿ)^2/n

Since the solutions to these equations are in rational numbers, which are quite complicated to solve further. Andrew Wiles who was an English student was interested in the theorem and gave proof of the Shimura-Taniyama-Weil conjecture. There was an error in the proof, but with the help of his student named Richard Taylor, he formulated a proof of Fermat’s theorem. Some of the proofs given by him were difficult and complex to understand. On October 6, he gave new proof to his colleagues which they found simple in comparison to previous ones. The proof stated by Andrew was published in the paper ‘Annals of Mathematics’ in 1995. However, some mathematicians still believe that there is no guarantee that the proof is completely accurate, and there always remains some doubt. 

Final Thoughts

The Fermat equation was solved by the mathematician himself that solved the case for n=4 effectively. With the help of computers, the theorem statement was confirmed by 1993 for all prime numbers less than 4,000,000. With the increasing time, mathematicians discovered that proving a special case of a result from number theory and algebraic geometry would be equivalent to giving Fermat’s last theorem proof.