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The Methods of Solving Equations Ax+By=Cz 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

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Abstract:

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 Ax+By=Cz 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.

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Periodical:
The Bulletin of Society for Mathematical Services and Standards (Volume 8)
Pages:
17-25
Citation:
K. R. R. Gandhi et al., "The Methods of Solving Equations Ax+By=Cz 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", The Bulletin of Society for Mathematical Services and Standards, Vol. 8, pp. 17-25, 2013
Online since:
December 2013
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References:

W. Sierpiński, On solution of equations in integers, (Russian), Fizmatgiz, Moscow, 1961. INTERNET LIBRARY: http: /ilib. mccme. ru/djvu/serp-int_eq. htm.

PROF. DR. K. RAJA RAMA GANDHI, REUVEN TINT, THE PROOF OF THE INSOLUBILITY IN NATURAL NUMBERS FOR THE FERMAT'S LAST THEOREM AND BEAL'S CONJECTURE FOR CO-PRIME INTEGERS ARRANGED IN A PAIR IN THE EQUATIONS, (ELEMENTARY ASPECT) 2013 Bulletin of Mathematical Sciences & Applications ISSN: 2278-9634 Vol. 2 No. 3.

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