It is well known, [1], that x and y cannot both be even numbers, and that they must be of different parity and relatively prime. Also, it is well known, [1], [2], that if the Last Theorem can be proved for n = 4, then it is also proven for all multiples of n = 4. Consequently, because all of the remaining numbers can be reduced to a multiple of the prime numbers, it is therefore only necessary to prove Fermat's Last theorem for all the primes. Indeed this is the approach adopted by many eminent mathematicians in the past, and more recently using computer analysis which, prior to Wiles' proof, established the proof of the Conjecture for all values of n up to 4 million, including the primes therein. However, it is obvious that this eventually becomes a fruitless task because, even if only prime numbers are considered, because there are an infinite number of prime numbers, the Conjecture can never be proven by such number crunching alone.

The method adopted here will however, consider such specific cases of n, because the results so obtained will be shown to exhibit a simple pattern, that can justifiably be extrapolated to the n^th order, so proving the general case. This method is initiated as follows.

Because x, y and z are simply numbers, with no physical meaning, ( 2.1) can be re-written as follows

Expanding ( 2.2) binomially and gathering all terms to the left gives

Application of Descartes' Rule of Signs shows that ( 2.3) contains just one positive root, with all the rest being negative. Also, from the co-efficient of x^{( n-1 )} it is clear that all the roots must contain the unique term (a - b). Furthermore, the nature of the other co-efficients also shows that the remaining roots cannot all be real. When n is even, in addition to the one positive root, there can only be one real negative root with all the rest being conjugate complex with negative real parts. And when n is odd, in addition to the one real positive root, all the rest are also conjugate complex with negative real parts.

Fermat's Last Theorem can therefore be proved by determining whether the one generalised positive real root of ( 2.3) can contain an integer value of the parameter a. The approach here is to accomplish this task for n = 3, 4, 5 and 7, from which extrapolation to the general case will be effected.