Natural Number System and Its Fundamental Relationships (Proof of Fermat's Last Theorem)

Pierre de Fermat in 17th century wrote as a marginal note (later published by his son Samuel de Fermat) on the Diophantus? book Arithmetica (Latin translation with commentary of the Greek book by Claude Gaspard de Bachet), while studying the natural number solutions of equation x2 + y2 =z2, that, ?No cubes of natural numbers can be split in to two cubes or a biquadrate can be split into two biquadrates or no other higher order greater than 2 of a natural number can be split in to the sum of two natural numbers having the same order, in which I have found a marvellous demonstration that this margin is narrow to contain.? Ironically, Fermat didn?t give the proof for this proposition during his life time. It was the last one to be proved of Fermat's propositions and hence historically called as Fermat's Last Theorem. In modern terms the theorem can be stated as ?xn +yn =zn has no solutions for n greater than2 in natural numbers.? Mathematicians tried for centuries but could not construct the proof for general case. In 1994 Prof. Andrew Wiles (with the help of Richard Taylor) published a proof using the advanced ideas and techniques of mathematics and is significantly long and deep. But the attempt here is to understand the fundamental relationships of natural number system, the linear and trilinear (triangle inequality) relationships in which the number system manifests its significance in physical world since the time of early civilizations. Fermat's Last Theorem is here by demonstrated as a statement about the uniqueness of the trilinear relationship in natural number system by establishing the unique and comprehensive correlation between Euclidean geometry and natural number system (geometric algebra of natural number system). Both of them in conjunction demonstrates the principle of true model of relational dominance of trilinear relationships in natural number system, in which the triangle law of addition of physical quantities and the trigonometry of the space fundamentally depends upon.