Fermat’s Last Theorem

Presenter: Hanh Than

FLT video http://www.youtube.com/watch?v=SVXB5zuZRcM

Pierre de Fermat

Pierre de Fermat (17 August 1601– 12 January 1665):

a French lawyer and an amateur mathematician.

Diophantine equation a Diophantine equation is an indeterminate polynomial

equation that allows the variables to be integers only.

Example:A linear Diophantine equation with two variables x and y: ax + by = c ( where a, b, and c are integers)

Pythagora’s theorem

Theorem: In any right triangle, the area of the square whose side is

the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

Pythagorean’s triple In Book II, Problem 8 of the Arithmetica, Diophantus poses the

problem of how to divide a given square number into the sum of two smaller squares.

In other words, solve the problem:

x2 + y2 = z2.

Any three numbers that satisfy this equation are called Pythagorean Triples.

Pythagorean triples (x, y, z) is primitive if x, y, z are pairwise co-prime.

Fermat’s Last Theorem (FLT) Fermat’s last theorem:

There is no non-zero integer solutions for all n > 2 that satisfy the equation

What did Fermat prove for FLT?

Fermat proved his Last Theorem for n = 4, using the method called "infinite descent" to prove that there are no positive integers, x, y, and z such that x4 + y4 = z4.

Moreover, if a solution exists for some n, the same solution also works for any multiple of n. Hence, only prime numbers have to be considered. Fermat also proved the theorem for n = 3.

Fermat: Method of infinite descent

Example: Show that there is no Pythagorean triple (a, b, c) with a = b

Did Fermat possess a general proof ?

Fermat wrote in the margin of bachet's translation of Diophantus's Arithmetica

“… I have discovered a truly remarkable proof which this margin is too small to contain. “

Brief history Fermat showed the case n= 3 and n = 4. Leonhard Euler showed independently for n = 3 and n = 4. 1816: The French Academy prize was announced. 1820’s: Sophie Germain showed that if p and 2p+1 are prime, then

xp + yp = zp has no solution with p does not divide xyz (case 1). 1825: Dicrichlet proved for n = 5. 1839: Lame’ proved for n=7. His proof for general n was failed

and it was pointed by Joseph Liouville. 1844-1847: Kummer worked on FLT. 1908: The Wolfskehl prize was offered for a solution for FLT. etc.,

Conjectures imply FLT By the late 1980’s, there were many conjectures in number

theory which, if proved, would imply FLT :

The abc conjecture.

Elliptic curves -- Taniyama-Shimura conjecture.

abc conjecture

The abc conjecture states that:

if there are three positive integers a, b, and c which share no common factor, that satisfy a + b = c , then the product of distinct prime factor is rarely much smaller than c

Elliptic curves General equation of elliptic curve over Q ( rational numbers):

y2= Ax3 + Bx2 + Cx + D ( where A, B, C, D are rational numbers and the cubic polynomial in x has distinct roots).

Elliptic curve & Taniyama-Shimura conjecture

Yukata Taniyama (1927– 1958): a Japanese mathematician.

Goro Shimura (1930 – present): a Japanese mathematician and currently a professor of mathematics at Princeton University.

Elliptic curve & Taniyama-Shimura conjecture

Taniyama-Shimura conjecture: Any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve.

Frey curves: a bridge between FLT and the Taniyama-Shimura conjecture.

How does FLT and Taniyama – Shimura conjecture link together?

Gerhard Frey (1944 - present): a Germany mathematician.

Frey curve: Frey showed that nontrivial solutions to FLT give rise to a special elliptic curves, called Frey curves.

That means if the Taniyama-Shimura conjecture were true, then Frey curves could not exist and FLT would follow.

Frey curves If ap + bp = cp is a solution to FLT, then the associated Frey

curve is:

y2 = x( x – ap )( x + bp) ( a, b, c are non-zero relatively prime integers and p is an odd prime)

Kenneth Alan Ribet: an American mathematician, and a professor at University of California, Berkeley.

In 1986, Ribet proved that Frey curve was not modular.

Taniyama-Shimura-Wiles

Andrew John Wiles (1953 – present): a British mathematician, and currently a professor of mathematics at Princeton University.

1986-1994:Wiles proved FLT indirectly by proving Taniyama-Shimura conjecture.

Summary

Taniyama-Shimura conjecture FLT was proved

Fermat’s last theorem

Did Fermat have FLT’s proof ?

Few mathematicians said YES

Some mathematicians said NO.

Question time