Fermat's Last Theorem

What is Fermat's Last Theorem?

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. 

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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, (zn/2 + yn/2 ) (zn/2 - yn/2) = xn.

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

zn/2 + yn/2 = 2n-1 pn

zn/2 - yn/2 = 2 qn

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

zn/2 = 2n-2 pn + qn

yn/2 = 2n-2 pn - qn

It gives:

z = (2n-2 pn + qn)2/n

y = (2n-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. 

FAQs (Frequently Asked Questions)

1. What are False Proofs of the Last Theorem Claimed by Fermat?

Throughout history, there have been several false proofs of the theorem last stated by Fermat. Many of them are proof for certain values of natural number n. These proofs satisfy only some values for the constants x, y, and z. Some proofs of Fermat's last theorem involve a lot of inequalities, Pythagoras Theorem, and trigonometry. These proofs are correct for selected even exponent and do not satisfy the equation for odd exponent. There also exist some near-miss solutions, also known as counterexamples to the Last Theorem stated by Fermat. Since some calculators cannot handle the arithmetic operations magnitude, so some integers satisfy the theorem-proof. One of the proofs includes showing that the equation does not hold for a large number of combinations rather concludes that it does not hold for all results. The proof shows only some finite number of likelihoods instead of showing all possibilities, as stated by the Fermat theorem.

2. Who Finalized the Fermats Last Theorem Proof?

Many mathematicians tried to prove the statement claimed by Fermat known as Fermat's Last Theorem. A counterexample to this theory was proposed by Andrew Wiles, who gave a lecture on Shimura-Taniyama-Weil at the Isaac Newton Institute, England. In the final lecture, Wiles announced the proof of the last theorem as a corollary to his main result. The mathematician has proved the Shimura-Taniyama-Weil conjecture for multiple examples. Among those examples, some were efficient to prove the last theorem. In 1993, the mathematician made a statement in which he stated that while reviewing the process, some of the problems had emerged and some of them were resolved. However, the final calculation was not yet completed, and he believed that soon he would be able to finish it. Later, Wiles sent the new proof to his colleagues and they found it much simpler compared to previous ones.