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Abstracts

XX conference

Second solutions of some whole number equations

Smolygin V.D.

Tel.: 8-916-834-87-96, E-mail; smolyg@yandex.ru

2 pp. (accepted)

SECOND SOLUTIONS OF SOME WHOLE NUMBER EQUATIONS

The possibilities have been proved for the use of full-parallel solution method (FP method) for finding the second nontrivial solutions for some whole number equations.

The second nontrivial solutions have been defined for the equation X^2+Y^2=Z^2 and equation x^2-Ay^2=1 in whole numbers.

For example, for the equation X^2+Y^2=Z^2:

- there are solutions: X_1=m^2-n^2, Y_1=2m∙n, Z_1=m^2-n^2 [1];

- other solutions: X_2=(2n-m)^2-n^2, Y_2=2(2n-m)∙n, Z_2=(2n-m)^2+n^2 [2].

Here: m and n are mutually heterogeneous prime * numbers, m>n.

The transformation of equations from multiple unknowns using FP method will define so many equations from one unknown as the number of unknowns contained in this equation; the highest degree of equation from one unknown is equal to the highest degree of monomials included into the equation from multiple unknowns.

Two solutions are defined for the equation (X+Y)^2=Z^2 in whole numbers.

Three solutions are defined for the equation (X+Y)^3=Z^3 in whole numbers.

The second solution examples are given for some of the solved equations in square whole numbers and higher from multiple unknowns.

* Two numbers one of which is even and another is odd are called heterogeneous.

References.

1. G.Rademacher and O. Teplits Numbers and figures. Experiments of mathematic thinking. M.: State publishing house of physical-mathematical literature, 1962. 264 pages.

2. Smolygin V.D. Two roots of equation type X^2+Y^2=Z^2 (Two solutions of the equation type X^2+Y^2=Z^2) // United scientific journal № 28. Moscow: Scientific publications’ fund. 2005. pg. 68-76.



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