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 there are no three positive whole numbers—a, b, and c—that can satisfy the equation $a^n + b^n = c^n$ for any integer value of n greater than 2. This famous conjecture puzzled mathematicians for centuries.
2. Who proved Fermat's Last Theorem?
The mathematician Andrew Wiles proved Fermat's Last Theorem in 1994. Wiles used advanced mathematical tools from algebraic geometry and modular forms, solving a problem that had remained unsolved for over 350 years.
3. Why is Fermat's Last Theorem important in mathematics?
Fermat's Last Theorem is important because it inspired new branches of mathematics, such as algebraic number theory. Its proof connected different areas, showing how ideas like
- modular forms
- elliptic curves
4. What is the equation in Fermat's Last Theorem?
The main equation for Fermat's Last Theorem is $a^n + b^n = c^n$, where
- a, b, and c are whole numbers greater than zero
- n is an integer greater than 2
5. How did Andrew Wiles prove Fermat’s Last Theorem?
Andrew Wiles proved Fermat's Last Theorem by showing that a specific type of mathematical object, called a semistable elliptic curve, is always modular. This proof used tools from
- number theory
- elliptic curves
6. Did Fermat really have a proof for his Last Theorem?
Pierre de Fermat claimed he had a marvelous proof for his theorem, but most historians and mathematicians now think he did not have a valid proof for Fermat's Last Theorem beyond the cases for n=3 and n=4.
7. What is an example that shows Fermat's Last Theorem is true?
In Fermat's Last Theorem, you cannot find positive whole numbers a, b, c where $a^3 + b^3 = c^3$. For instance, 3, 4, and 5 do not satisfy $3^3 + 4^3 = 5^3$ because 27 + 64 ≠ 125.
8. What does "n" mean in Fermat's Last Theorem?
In Fermat’s Last Theorem, n is an integer greater than 2. It represents the exponent in the equation $a^n + b^n = c^n$. The theorem deals with the impossibility of solutions for these values of n using positive whole numbers only.
9. Which mathematical fields are related to Fermat’s Last Theorem?
Fields related to Fermat's Last Theorem include
- number theory
- modular forms
- algebraic geometry
10. What are some special cases of Fermat's Last Theorem?
Special cases include when n = 3 or n = 4. Mathematicians solved these specific exponents before the general case. For $n = 3$, Euler proved it, while
- Fermat himself proved it for n = 4
11. How long did it take to prove Fermat’s Last Theorem?
Fermat’s Last Theorem was stated in 1637, but a complete proof was not found until 1994 by Andrew Wiles. This means the theorem remained unproven for over 350 years, making it one of the longest-standing unsolved math problems.
