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Diophantine Equations
•
These are polynomial equations for which postive integer solutions are sought.
•
Diophantus’s
Arithmetica
(250) consisted of solutions to one or two hundred Diophantine
equations, but neither complete solution sets nor systematic solution techniques were pre
sented.
•
In 1900, David Hilbert gave a list of 23 unsolved math problems he thought were especially
important. These are referred to as “Hilbert’s Problems”.
•
Hilbert’s 10th Problem was to come up with a general algorithm for determining whether a
Diophantine equation has a solution.
•
In 1970 Yuri Matiyasevich proved that there is no such algorithm.
Fermat’s Last Theorem
•
The Theorem: If
n >
2 there are no numbers
x
,
y
, and
z
such that
x
n
+
y
n
=
z
n
.
•
Fermat wrote in the margin of his copy of Diophantus’
Arithmetica
: “It is impossible to
separate a cube into two cubes, or a fourth power into two fourth powers, or in general,
any power higher than the second into two like powers. I have discovered a truly marvelous
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 Fall '10
 wen
 Equations, Sets

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