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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.  

Since you asked us this question, your teacher may have told you about 
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
Associated Topics:
High School Geometry
High School Number Theory
High School Triangles and Other Polygons

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