A DEMONSRATIONEN MIRABILEN OF FERMAT'S LAST THEOREM
MANGATIANA A. ROBDERA
Abstract. We give a simple direct proof of the Fermat's last theorem.
Given a positive integer n, a triple (x, y, z) of positive integers satisfying the equation xn + y n = z n is called a Pythagorean triple of degree n. The Fermat's Last Theorem (FLT) states that Theorem 1.
There is no Pythagorean triple of degree
n > 2.
We observes that since all positive integers can be reduced to a multiple of prime numbers, it is necessary to prove FLT for all odd prime numbers n. We also observe that if (x, y, z) is a Pythagorean triple and if d > 1 divides each of x, y, z , then xd , yd , dz is also a Pythagorean triple. In what follows, we x an odd prime number n and a relatively prime Pythagorean triple (x, y, z) of degree n. We assume that x < y and let h = z − x. We note that the rst nth -power integer after xn is (x + 1)n . Thus if xn + y n were to be an nth -power integer n then one must have (x + 1)n P < xn + yn = (x + h)n < P (x + y) , and thus 1 < h < y . Clearly, xn + y n = z n is n n n n−p p n n−p p n n n n equivalent to x + y = x + p=1 p x h or y = p=1 p x h . Thus y n , and hence y , is divisible by h. Thus y n is divisible by hn , (n > 2). On the other hand, we have Pn • either h 6= n; in which case y n = nxn−1 h + p=2 np xn−p hp is not divisible by h2 , a contradiction; Pn • or h = n; in which case, y n = n2 xn−1 + p=2 np xn−p np is divisible by n2 = h2 but not by n3 = h3 ; another contradiction. The proof is complete. Proof.
References [1] Wiles, A.,
Modular elliptic curves and Fermat's last theorem. Annals of Mathematics. 1995;141(3):443-551.
Department of Mathematics, University of Botswana, 4775 Notwane Road, Gaborone, Botswana
Email address :
[email protected]
2010 Mathematics Subject Classication. 11A-99. Key words and phrases. Fermat's last theorem.
1