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Search: id:A131386
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| A131386 |
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We start from a generalized Diophantine Equation : Z^n=X_1^{n_1}+...X_i^{n_i} n_j , n , X_j, Z are positive integers, X_j, Z are coprime. For ,i=2, n_j=n it is Fermat equation. For i=2 , it is Fermat-Catalan (or Beal). After a little change of the data, we define the following sequences (articles published by the Asian Journal of Algebra, copyright) x_i= rac{x^{2^{i-1}}}{x^{2^{i-1}}-y^{2^{i-1}}}(x-y) y_i= rac{y^{2^{i-1}}}{x^{2^{i-1}}-y^{2^{i-1}}}(x-y) z_i=x_i+y_i u_i= rac{x_iy_i}{x_i+y_i} The coefficients of z_i in function of z_{i-1} and u_{i-1} beginning from i=3 . The sequence is, then 1, -2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,… (We proved that x_i-y_i=x-y=0 ). |
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1
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| 1, -2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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This sequence is generated by a generalized Diophantine Equation. It has no formula, but it seems that a(2k+1)=1 for all k>0 and a(2)=-2, a(2k)=0 for all k>1.
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REFERENCES
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A. D. Aczel, Fermat's Last Theorem, Four Walls Eight Windows NY 1996
A. C. Clarke, The Last Theorem, Gollancz SF 2004.
B. Cipra, What's Happening in the Mathematical Sciences 1994 Vol. 2, "A Truly Remarkable Proof" pp. 3-8 AMS Providence RI.
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LINKS
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Article number 1
Article number 2http://www.ansijournals.com/aja/2008/15-24.pdf
A short form proof of FLT
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FORMULA
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a(1)=1, a(2)=-2, a(2k+1)=1, a(2k)=0, k\geq{1}.
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EXAMPLE
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We calculate x_2, y_2, z_2, u_2, and x_3, y_3, z_3, u_3, we discover (tere is no formula) a(1)=1, a(2)=-2, and x_4, y_4, z_4, u_4, we discover a(3), a(4), etc...
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KEYWORD
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easy,nonn,new
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AUTHOR
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Jamel Ghanouchi (jamel.ghanouchi(AT)topnet.tn), Aug 26 2008
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