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Theorem 1. There is no Pythagorean triple of degree n > 2. Proof. We observes that since all positive integers can be reduced to a multiple of prime numbers, it is ...
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.

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