There was a necessary restructuring of the references and corresponding changes in the main body of the first version of the paper. This has been corrected in the following version.
Ever since the French mathematician Pierre de Fermat  stated the conjecture in 1637 that the equation cannot have any solutions if a, b, and c are integers and n is an integer , the equation has been a subject of intense and often heated discussion amongst mathematicians and non-mathematicians alike. This conjecture is known as Fermat’s Last Theorem. The fact that Fermat claimed to have a proof but never wrote it down has put researchers in a quandary since nobody has yet been able to duplicate the proof the way Fermat originally claimed. Perhaps the strong appeal of the problem is the simplicity and elegance of its statement contrasted with the apparent hopelessness  of finding an elementary way to establish it. Finally, around 1994 based on a property called modularity of an elliptic curve, Andrew Wiles [11, 10] offered a proof of the theorem. His paper incorporates by reference [7, 3] a vastly larger body of mathematical work developed over the last several decades. But it requires an extraordinary arsenal  of mathematical tools to understand Wiles’s complex and very lengthy proof. The proof vastly differs in scope and complexity from the proof Fermat originally envisioned. Consequently, the quest for a simple and short proof continues. Of course any general proof of the theorem will also imply the proof of any special case. In this paper, based on elementary principles, a simple proof of the theorem is given for even exponents .
Equation (1) has no nonzero integer solutions when the exponent n is an integer :
Simplification of the theorem.
Any integer is either divisible by 4 or by an odd prime. Fermat’s Last Theorem is already known  to be true when n is a multiple of 3 or 4. Again, x, y, and z must not have any common factor. Otherwise, both sides of the equation can be divided by the common factor to obtain a smaller solution. Also for consistency only one of the variables can be even. When z is even, the left-hand side is equivalent to 2 (mod 4) and the right-hand side is equivalent to 0 (mod 4). This leads to an inconsistency. Again since Fermat’s equation deals with the situation where all three variables have like powers, it is enough to prove the theorem when the three variables x, y, z are relatively prime integers, y is even, the exponent n is a prime and none of the variables  is a prime.
Equation (1) has been of great interest to number theorists for a long time. In 1837, E. E. Kummer [2, 8] proved that if (1) has integer solutions then 1 (mod 8). Rothholtz  extended Kummer’s result to prove that (1) has no integer solution if the exponent n is a prime of the form or one of the variables x, y, z is a prime. In 1977 Terjanian  offered a surprisingly simple proof that if (1) is satisfied for nonzero integers then n divides x or y. Equivalently, Terjanian proved Fermat’s Last Theorem for the first case with even exponents. In this paper a simple proof of the theorem is offered for all even exponents.
Search for integer solutions.
Throughout the paper all the variables are positive reals, means that x, y, z are coprime integers, and means that a, b are coprime integers and .
Fermat’s equation with an even exponent has integer solutions.
, Z is an odd square, X, Y are k-th powers, k is a prime . It is noted that (2) can have integer solutions as seen from the example . The objective here is to show that the solutions of (2) cannot be of the form , , where . Since integer solutions of (1) are assumed by using Terjanian’s result  one notes .
Under the assumptions z, g, and h are all integers , equation (9) represents a right triangle ZGH whose sides and area are integers, and z is the hypotenuse. Therefore, ZGH is a rational right triangle . Equivalently, is a Pythagorean triple. Consequently, equations (10), (11), (12), and (13) are obtained where , and :
where are nonzero integers, each divisible by k. Moreover, if (mod 4), and otherwise. The sign of s will influence only the orientation of X and Y but will have no impact on the integer solutions of (1). Also since .
Since and , one concludes that and . Therefore, if X and Y are k-th powers, then g, Q, h and R must assume values of the forms , , , and , where u, r, v, and d are integers , and w is an integer .
The impossibility of (22) will imply the impossibility of (1). By expanding in terms of (see [4, p. 111]), one gets (23), where U and V are given by and . It is noted that and . One thus gets (24) and (25):
Under the assumption equation (24) can not be satisfied.
where , , and . Here z is the hypotenuse of the right triangle ZGH, hence has linear dimension and is dimensionless. It is seen that the left-hand side of (26) has dimension k and the right-hand side has dimension . This leads to an inconsistency and justifies the assertion.
One numerical example.
The author wishes to thank the referee for his/her valuable comments on the initial version of the manuscript.
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