What Makes Fermat’s Last Theorem a Landmark in Mathematics?
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 on Fermat’s Last Theorem: Meaning, Proof & Importance
1. What is Fermat's Last Theorem?
Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of 'n' greater than 2. While the related Pythagorean theorem (a² + b² = c²) has countless integer solutions, Fermat's theorem asserts that as soon as the exponent becomes 3 or higher, no such integer solution exists.
2. Who proved Fermat's Last Theorem?
Fermat's Last Theorem was famously proven by the English mathematician Andrew Wiles in 1994. After discovering a flaw in his initial 1993 proof, he worked with his former student Richard Taylor to fix it. The final proof was a monumental achievement, connecting two distinct areas of mathematics and resolving a problem that had puzzled mathematicians for over 350 years.
3. Can you explain Fermat's Last Theorem in simple terms?
Imagine you have building blocks of different sizes. The Pythagorean theorem (a² + b² = c²) tells us you can find square-shaped areas that perfectly add up. For instance, the area of a 3x3 square plus a 4x4 square equals the area of a 5x5 square. Fermat's Last Theorem simply says that if you try to do the same with cubes (a³ + b³ = c³) or any higher-dimensional shapes (powers greater than 2), you will never find whole-numbered blocks that fit together perfectly.
4. Why is Fermat's Last Theorem considered so important in mathematics?
The importance of Fermat's Last Theorem lies not in the result itself, but in the journey to find its proof. Its significance comes from several factors:
Mathematical Innovation: The centuries-long quest for a proof led to the development of major new fields in number theory, such as algebraic number theory and the theory of elliptic curves.
The Taniyama-Shimura Conjecture: The final proof confirmed a profound connection between two seemingly unrelated mathematical worlds: elliptic curves and modular forms. Proving this conjecture was key to proving Fermat's Last Theorem.
A Symbol of Mathematical Pursuit: It represents a classic example of a simple-to-state problem that required incredibly deep and complex new ideas to solve, inspiring generations of mathematicians.
5. What is the connection between Fermat's Last Theorem and the Pythagorean Theorem?
Fermat's Last Theorem is a direct generalisation of the Pythagorean Theorem. The Pythagorean Theorem is the specific case of the equation aⁿ + bⁿ = cⁿ where the exponent n = 2. For n=2, there are infinite whole number solutions, known as Pythagorean triples (like 3, 4, 5). Fermat's Last Theorem explores what happens for all other integer exponents (n=3, n=4, etc.) and concludes that for any n > 2, there are zero whole number solutions.
6. How is Fermat's Last Theorem different from Fermat's Little Theorem?
Though both are named after Pierre de Fermat, they are very different concepts:
Fermat's Last Theorem is about the non-existence of integer solutions to the equation aⁿ + bⁿ = cⁿ for n>2. It was a famous unsolved problem for centuries.
Fermat's Little Theorem is a fundamental result in number theory concerning modular arithmetic. It states that if 'p' is a prime number, then for any integer 'a', the number aᵖ - a is an integer multiple of 'p'. It has a known proof and is widely used in applications like public-key cryptography.
7. Why did it take over 350 years to prove Fermat's Last Theorem?
The proof took so long primarily because the mathematical tools needed to solve it did not exist in Fermat's time. While Fermat may have had a proof for a specific case (like n=4), a general proof that worked for all exponents greater than 2 was extraordinarily difficult. The final proof by Andrew Wiles was over 100 pages long and relied on highly advanced 20th-century mathematics that connected different fields in a way no one had previously imagined, a feat far beyond the scope of classical number theory.
8. Does Fermat's Last Theorem have any real-world applications?
Fermat's Last Theorem itself does not have any direct practical or real-world applications. It is a statement of pure mathematics. However, the mathematical fields that were developed in the attempt to prove it are incredibly important. For example, concepts from algebraic number theory and the theory of modular forms (which were central to its proof) are now fundamental in modern cryptography, which secures internet communications, banking, and data privacy.
