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.
