A Proof of Last Fermat Theorem Abdelmajid Ben Hadj Salem, Dipl.-Eng. Email:
[email protected] 6, Rue du Nil, Cité Soliman Er-Riadh, 8020 Soliman, Tunisia. Abstract We give a proof of Pierre de Fermat’s last theorem using that Beal Conjecture is true.
Keywords: Prime numbers, divisibility, proof of Beal conjecture.
To my Wife Wahida
1
Introduction
In 1621, Pierre de Fermat published the following theorem (called the Fermat Last Theorem) : Theorem 1.1. There are no solutions of: Am + B m = C m
(1.1)
with A, B, C, m, be positive integers with m > 2. In this paper, we give a proof of this theorem using that Beal Conjecture [1] is true.
2
The Proof
Proof. We suppose that for m ∈ N, m > 2, there is a solution (A, B, C) ∈ N3 of (1.1). Using that Beal conjecture [2] is true, then A, B and C have a common factor. Let µ ∈ N be the great common factor that divides A, B, C. Then we write: A = µA1
(2.1)
B = µB1
(2.2)
C = µC1
(2.3)
with (A1 , B1 , C1 ) are co-prime. The equation (1.1) becomes: m m Am 1 + B1 = C 1
In the following, we suppose that A1 > B1 .
1
(2.4)
2 The Proof
2.1
2
B1 > 1
As m > 2, we use the Beal Conjecture, then A1 , B1 , C1 have a common factor >1 which is a contradiction with A1 , B1 , C1 co-prime. Then the equation (1.1) has no integer solutions.
2.2
B1 = 1
Then we obtain: m Am 1 + 1 = C1 ⇒ C1 > A1
(2.5)
We write C1 = A1 + c, c ∈ N, then : m (A1 + c)m = Am 1 +1⇒c +
k=m−1 X
k k m−k Cm A1 c =1
(2.6)
k=0
which is impossible. Q.E.D
References [1] R. Daniel Mauldin. A Generalization of Fermat’s last theorem: the Beal conjecture and prize problem. Notice of AMS, Vol 44, 1436–1437 (1997) [2] A. Ben Hadj Salem. A Complete Proof of Beal Conjecture. Paper submitted to the journal of the European Mathematical Society (JEMS). 64 pages. 2018. Published in /https : //www.researchgate.net/publication/323389472/.
