X^n + Y^n = Z^n if n is whole number and X,Y, and Z are whole numbers. Show that his is a false statement.
This is the original question posed by Fermat.
Let us show the statement is false for n=3 Thus, X^2 +Y^2 = h*Z^2 where h is some positive number greater than 1. Also, X +Y = l*Z where l is some positive number greater than 1. If we add 2XY to both sides of equation #2, the equation now becomes h*Z^2 +2X*Y = X^2 +2X*Y+Y^2. This now becomes h*Z^2 + 2X*Y = (X +Y)^2 = l*Z^2 Shuffling some terms you get (h- l^2)*Z^2=-2X*Y Further snuffling gives l^2= h-2X*Y/Z^2. Finally multiplying h by 2/2 we get l^2= 2h/2 -2X*Y/Z^2 Factoring the 2 outside the brackets we get l^2 = 2(h/2 -X*Y/Z^) So, l= (h/2 ? X*Y/Z^2) * 2^1/2 l is rational, as are X,Y,Z, and h. Here is the contradiction. The right hand side is an irrational number because 2^1/2 is a irrational number, while all the symbols inside the brackets rational. The right hand side of equation is therefore a irrational number. The left hand side is rational. The original statement is false then. Finally, since the proof can be applied for any n>2, this proofs the original assertion b Fermat.