Transcript Fermat`s last theorem - University of Toronto Mississauga

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Presenter: Hanh Than
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http://www.youtube.com/watch?v=SVXB5zuZRcM
Pierre de Fermat (17 August
1601– 12 January 1665):
a French lawyer and an
amateur mathematician.
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a Diophantine equation is an indeterminate polynomial
equation that allows the variables to be integers only.
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Example:
A linear Diophantine equation with two variables x and y:
ax + by = c ( where a, b, and c are integers)
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).
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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.
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Pythagorean triples (x, y, z) is primitive if x, y, z are pairwise coprime.
Fermat’s last theorem:
There is no non-zero integer solutions for all n > 2 that
satisfy the equation
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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.
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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.
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Example: Show that there is no Pythagorean triple (a, b, c)
with a = b
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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. “
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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.,
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By the late 1980’s, there were many conjectures in number
theory which, if proved, would imply FLT :
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The abc conjecture.
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Elliptic curves -- Taniyama-Shimura conjecture.
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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
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).
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Yukata Taniyama (1927– 1958): a
Japanese mathematician.
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Goro Shimura (1930 – present): a
Japanese mathematician and
currently a professor of
mathematics at Princeton
University.
Taniyama-Shimura conjecture:
Any elliptic curve over Q can be obtained via a rational map
with integer coefficients from the classical modular curve.
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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.
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That means if the Taniyama-Shimura conjecture were true, then Frey
curves could not exist and FLT would follow.
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)
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Kenneth Alan Ribet: an American
mathematician, and a professor at University
of California, Berkeley.
In 1986, Ribet proved that Frey curve was not
modular.
Andrew John Wiles (1953 – present):
a British mathematician, and currently a
professor of mathematics at Princeton
University.
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1986-1994:Wiles proved FLT indirectly
by proving Taniyama-Shimura conjecture.
Taniyama-Shimura conjecture
FLT was proved
Fermat’s last theorem
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Few mathematicians said YES
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Some mathematicians said NO.