example of Fermat’s last theorem
has no solution in positive integers.
where , are coprime and have opposite parity. Since is a primitive Pythagorean triple, we have coprime , of opposite parity satisfying
From it follows that , which implies . Since is a square, each of is a square.
Setting , , leads to
where . Thus, equation 3 gives a solution where . Applying the above steps repeatedly would produce an infinite sequence of positive integers, each of which was the sum of two fourth powers. But there cannot be infinitely many positive integers smaller than a given one; in particular this contradicts to the fact that there must exist a smallest for which (1) is solvable. So there are no solutions in positive integers for this equation. ∎
A consequence of the above theorem is that the area of a right triangle with integer sides is not a square; equivalently, a right triangle with rational sides has an area which is not the square of a rational.
|Title||example of Fermat’s last theorem|
|Date of creation||2013-03-22 14:09:51|
|Last modified on||2013-03-22 14:09:51|
|Owner||Thomas Heye (1234)|
|Last modified by||Thomas Heye (1234)|
|Author||Thomas Heye (1234)|