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We give the corresponding identities for different solutions of the equations: aA x +bB x =cD x  [1] and aA x +bB y =cD z  [2]: As for coprime integers a, b, c, A, B, D and arbitrary positive integers x, y, z further, for not coprime integers, if A 0  x 0  +B 0  x 0  =D 0  x o   [3] and A 0  x 0  +B 0  y o  =D 0  z 0   [4], where x 0 , y 0 , z 0 , A 0 , B 0 , D 0  - are any solutions in positive integers.

bulletin of mathematical sciences and applications

The Proof of the Insolubility in Natural Numbers for n&amp;gt;2, the Fermat's Last Theorem and Beal's Conjecture for Co-Prime Integers Arranged in a Pair A, B, D in the Equations A&lt;sup&gt;n&lt;/sup&gt;+B&lt;sup&gt;n&lt;/sup&gt;=D&lt;sup&gt;n&lt;/sup&gt; and A&lt;sup&gt;n&lt;/sup&gt;+B&lt;sup&gt;y&lt;/sup&gt;=D&lt;sup&gt;z&lt;/sup&gt; (Elementary Aspect)