x̄ - > Fermat's Last Theorem

 Fermat's Last Theorem is a significant mathematical proposition originally suggested by Fermat in a note written in the margins of a copy of the ancient Greek text Arithmetica by Diophantus. Although the original note is lost, a copy was preserved in a book published by Fermat's son after his death. In the note, Fermat claimed to have discovered a proof for the equation x^n+y^n=z^n, where n is greater than 2 and x, y, z are integers (excluding zero), showing that it has no integer solutions.

Due to Fermat's statement, this proposition became known as Fermat's Last Theorem, even though it remained unproven for many centuries. It should be noted that the restriction n>2 is necessary because there are formulas that generate infinite Pythagorean triples (x, y, z) satisfying the equation for n=2.

Various attempts were made to solve the equation, such as factoring it or exploring specific cases, but these approaches did not provide significant insights or solutions. The theorem can be divided into two cases: one where the exponent is relatively prime to x, y, and z, and another where the exponent divides exactly one of x, y, or z.

Over time, several mathematicians made progress on specific cases of the theorem. For example, Euler, Fermat himself for n=4, Dirichlet, Lagrange, and others proved the theorem for certain values of n. However, a general proof remained elusive.

In 1993, Andrew Wiles partially proved the theorem by establishing the semistable case of the Taniyama-Shimura conjecture. Although some flaws were found in the initial proof, Wiles and R. Taylor addressed these issues, and the complete proof was published in 1995. Wiles' approach involved using Galois representations and reducing the problem to a class number formula.

Wiles' proof marked a significant milestone in mathematics, as it brought an end to a long-standing problem. It is interesting to consider whether Fermat himself had an elementary proof, but given the difficulty of the theorem and the lack of tools available during his time, it is likely that his alleged proof was not valid.

Fermat's Last Theorem has also been referenced in popular culture, such as appearing in an episode of "The Simpsons" and being mentioned in an episode of "Star Trek: The Next Generation."

Fermat's last theorem is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The scribbled note was discovered posthumously, and the original is now lost. However, a copy was preserved in a book published by Fermat's son. In the note, Fermat claimed to have discovered a proof that the Diophantine equation x^n+y^n=z^n has no integer solutions for n>2 and x,y,z!=0. The full text of Fermat's statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain." As a result of Fermat's marginal note, the proposition that the Diophantine equation x^n+y^n=z^n, (1) where x, y, z, and n are integers, has no nonzero solutions for n>2 has come to be known as Fermat's Last Theorem. It was called a "theorem" on the strength of Fermat's statement, despite the fact that no other mathematician was able to prove it for hundreds of years. Note that the restriction n>2 is obviously necessary since there are a number of elementary formulas for generating an infinite number of Pythagorean triples (x,y,z) satisfying the equation for n=2, x^2+y^2=z^2. (2) A first attempt to solve the equation can be made by attempting to factor the equation, giving (z^(n/2)+y^(n/2))(z^(n/2)-y^(n/2))=x^n. (3) Since the product is an exact power, {z^(n/2)+y^(n/2)=2^(n-1)p^n; z^(n/2)-y^(n/2)=2q^nor{z^(n/2)+y^(n/2)=2p^n; z^(n/2)-y^(n/2)=2^(n-1)q^n. (4) Solving for y and z gives {z^(n/2)=2^(n-2)p^n+q^n; y^(n/2)=2^(n-2)p^n-q^nor{z^(n/2)=p^n+2^(n-2)q^n; y^(n/2)=p^n-2^(n-2)q^n, (5) which give {z=(2^(n-2)p^n+q^n)^(2/n); y=(2^(n-2)p^n-q^n)^(2/n)or{z=(p^n+2^(n-2)q^n)^(2/n); y=(p^n-2^(n-2)q^n)^(2/n). (6) However, since solutions to these equations in rational numbers are no easier to find than solutions to the original equation, this approach unfortunately does not provide any additional insight. If an odd prime p divides n, then the reduction (x^m)^p+(y^m)^p=(z^m)^p (7) can be made, so redefining the arguments gives x^p+y^p=z^p. (8) If no odd prime divides n, then n is a power of 2, so 4|n and, in this case, equations (7) and (8) work with 4 in place of p. Since the case n=4 was proved by Fermat to have no solutions, it is sufficient to prove Fermat's last theorem by considering odd prime powers only. Similarly, is sufficient to prove Fermat's last theorem by considering only relatively prime x, y, and z, since each term in equation (1) can then be divided by GCD(x,y,z)^n, where GCD(x,y,z) is the greatest common divisor. The so-called "first case" of the theorem is for exponents which are relatively prime to x, y, and z (px,y,z) and was considered by Wieferich. Sophie Germain proved the first case of Fermat's Last Theorem for any odd prime p when 2p+1 is also a prime. Legendre subsequently proved that if p is a prime such that 4p+1, 8p+1, 10p+1, 14p+1, or 16p+1 is also a prime, then the first case of Fermat's Last Theorem holds for p. 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