An Elementary Proof of Fermat’s Last Theorem for Epsilons

The author presents a new approach which is used to solve an important Diophantine problem. An elementary argument is used to furnish another fully transparent proof of Fermat’s Last Theorem. This was first stated by Pierre de Fermat in the seventeenth century. It is widely regarded that no elementary proof of this theorem exists. The author provides evidence to dispel this belief.


Introduction
Define n to be any integer such that (1). A brief history of the subject is given in [8]. A new proof showing that (1) is insoluble for 2 n > is provided in the subsequent section. The author has previously discovered another simpler proof [9] of Fermat's Last Theorem by deriving the fact that (1) has no solutions for n c ≥ . A completely different approach is used here. The following lemmas are fundamental results that are necessary for the novel argument employed in the proof of the theorem that follows.
Note that the trivial case b = c is discarded by considering Lemma 3.

Analysis
Lemma 1. If (1) holds only for 2 n > , then it may be assumed without loss of generality that n is an odd prime.
Proof. Suppose that 2 n p m < = ⋅ , where p is prime and m is some positive integer. It is possible to rewrite (1) as Suppose that q is some odd prime. The integer n, being at least 3, is divisible by an integer y such that . It is proved in [1] that (1) The statement of the lemma follows immediately. □ Lemma 3. Suppose that (1) is true. Then it may be assumed without loss of generality that , , a b c are pairwise coprime. Proof. Divide , , n n n a b c in (1) by their greatest common divisor. The statement of the lemma follows immediately. □ The following theorem is an important result which is established in an original and simple manner.
Theorem 1. Suppose that 2 n > . Then (1) has no solution. Proof. Suppose that (1) holds. By considering Lemma 1, it may be assumed without loss of generality that n is an odd prime. Since n is odd, By considering Lemma 2 with the fact that 0 a b c > > > , it can be deter- , so that there exists some integer a p which is a common factor of a and ( ) b c + such that 1 a p > . The last three pairs of equations can be used to distinguish various cases.
, it may then be assumed without loss of generality in the subsequent argument that a p can be chosen such that By considering the last two equations with (5), it can be established that   (6) and (7) (14) and (16)

Conclusion and Discussion
A Diophantine problem known as Fermat's Last Theorem has been solved by using a new elementary proof by contradiction. It was motivated by considering factoring an equation with odd exponents. The method of proof involved analyzes three pairs of cases before using them to formulate a novel proof by contradiction. This method is more economical than using more advanced techniques to prove the desired result that the original Diophantine equation has no solutions for 2 n > . Despite several attempts to obtain readers spanning the course of almost a year, the author could not find anyone who was prepared to properly read and check this proof of Fermat's Last Theorem. As the author has not been aware of any possible mistakes in the proof before publication, a decision has been made to publish this paper in case it may be noticed. Only in this instance can the proof be properly checked. It is mentioned here that the author has established Fermat's Last Theorem in a completely different manner in [9].
In this paper, the author has attempted to supply another elementary proof which is intended to be of a more conventional nature than the proof in [9].

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.