In this article the author introduces the notion of the binary mathematical statement A n from natural parameter n and refined axiomatic Peano natural numbers by adding the axiom of descent which is algebraic interpretation of the so-called method of descent Fermat. The known class of the Diophantine equations Fermat is reduced to some class of the algebraic equations from natural parameter n, n ≥ 3 (degree of polynomial). It is proved that concerning binary statement B n: ”whether has the equation for a preset value n some decision x n” the corresponding classes of the algebraic and Diophantine equations are equivalent. We show that the constructed class of the algebraic equations has rational decision only for n = 4. For n = 3, 4 the constructed class of the algebraic equations has decisions in radicals, and for n ≥ 5 this classes of the equations isn’t solvable at all. Thus also it is prove, that the Great Hypothesis of Fermat is correct with the small precision: the class Diophantine equations of Fermat have not the decision non-only in integers but and in the rational field.
Binary mathematical statement, axiom of descent, algebraic equation, Diophantine equation.