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Topic: Fermat's last Theorem in 13 statementws
Replies: 0

 Michael C Weir Posts: 1 Registered: 5/1/13
Fermat's last Theorem in 13 statementws
Posted: May 1, 2013 11:26 PM

Fermat's last Theorem

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.