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### Pythagorean Theorem, Fermat's Last Theorem

```
Date: 5/16/96 at 11:52:4
From: Anonymous
Subject: Geometry

I have been challenged by a teacher to see if a cubed + b cubed = c
cubed.   We are doing the Pythagorean theorem.  My question is, can
the equation be done with 3 different numbers.  It might be a trick
question.  Thanks.
```

```
Date: 7/12/96 at 12:8:59
From: Doctor Beth
Subject: Re: Geometry

Hi!  If I understand correctly, you want to know if a^3 + b^3 = c^3
for 3 different positive integers (none of them 0).  This is a famous
theorem in mathematics, and it was proven about a year or so ago by
Andrew Wiles that it cannot be done.

Fermat's Last Theorem (which Fermat never proved) that a^n + b^n = c^n
has no solutions of 3 distinct positive integers when n is also a
positive integer that's larger than 2.  Andrew Wiles' proof of this
theorem uses some really advanced mathematics that even most
professional mathematicians don't understand!

If you wanted integers to solve a^3 + b^3 = c^3 and didn't require
that they be positive, then you could let a = -1, b = 1 and c = 0.
You also know that there are lots of numbers that solve the equation
a^2 + b^2 = c^2, like a=3, b=4, and c=5, or also a = 5, b = 12, and
c = 13.

-Doctor Beth,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Number Theory
High School Triangles and Other Polygons

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