**Authors:** Vladimir Godovalov

Nature does not create anything extra. Mathematics as a part of Nature obeys the same law. If figurate numbers exist in nature, it means there is reason for their existence. In fact they represent a key to many solutions and serve as a foundation in Powers Fields Theory. The theory opens a new chapter in mathematics, which studies interaction of monomials n^m in homogenous areas called the powers fields. More precisely, these areas consist of consecutive and interconnected values organized in rows by their common attribute – the exponent m itself. The theory is based on Monomial Decomposition Theorem which firstly leads to structural organization of said areas, secondly, settles the powers field’s basic equations and finally allows the areas elements be expressed as figurative and factorial polynomials. Because of that the nature of equation a^x+b^y=c^z becomes a not complicated subject to systematic analysis enabling the theory to reveal in detail its matter. Technically the analysis falls in two ways. It begins with analysis of the powers field’s properties, the first of which actually states the Fermat’s conjecture. Being in fact not an independent problem by itself, Fermat’s conjecture is a technique applied in studying of the powers fields. Other powers field’s property, currently unknown to modern mathematics, is based on the genus-structural properties of figurative polynomials and therefore carries the same name. And finally, performing the most extensive study, the Beal’s conjecture ends analysis by searching solutions to the equations as well as determines among them cases with common prime factor. In addition to studying the powers fields as a main objective, the theory introduces several innovative methods along with new functions and definitions. Also, the paper includes Composite Numbers Theorem, the proven results of which are well known in modern mathematics formulas of factoring sum/difference of n-powers. The theory not only does discover many links within modern mathematics, it also raises a set of new questions in more specific areas of the theory and one of them for example is Phantom problem. The emergence of Powers Fields Theory not only fills a gap in Number Theory, but also sheds light on many related issues.

**Comments:** 109 Pages. Russian version

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[v1] 2012-06-07 03:50:33

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