The Methods of Solving Equations A^x+B^y=C^z with Co-Prime A, B, C, where x≥2,y≥2,x≥2 Are Natural Numbers, Equal the Two only in one of the Three Possible Cases - The Proof of Catalan's Conjecture

One of the principal problems of the Beal's conjecture, as we see that, is methods for finding a pairwise coprime solution which is defined below. First found methods and identities, allowing receiving infinite number solutions of equations as A^x+B^y=C^z for co-prime integers arranged in a pair (A,B,C)=1 are natural (whole) numbers, where a fixed permutation (x,y,z)corresponds to each of the permutations (2,3,4), (2,4,3), (4,3,2) Here we obtain also our method and identities of all not recurrent and not co-prime solutions of the above type, part of which has already been published, in contrast to the method of obtaining the recurrence not co-prime solutions of this type from [(1), W. Sierpiński, p. 21-25, 63]. As the solution of the main problem appeared additional problems that solved by obtained appropriate identities. Given as two equal proofs of Catalan's Conjecture.