> Although all issues related to the resolution of Fermat’s last theorem have been fully debated worldwide since 1997 and NOTHING had been conceded from my side I have seen at least one post expressing some misunderstanding. Let me, therefore, make the following clarification:

> 1) The decimal integers N.99… , N = 0, 1, …, are well-defined nonterminating decimals among the new real numbers [8] and are isomorphic to the ordinary integers, i.e., integral parts of the decimals, under the mapping, d* -> 0, N+1 -> N.99… Therefore, the decimal integers are integers [3]. The kernel of this isomorphism is (d*,1) and its image is (0,0.99…). Therefore, (d*)^n = d* since 0^n = 0 and (0.99…)^n = 0.99… since 1^n = 1 for any integer n > 2.  

> 2) From the definition of d* [8], N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that  ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … [8].

> 3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
> x^n + y^n = z^n,                                                                                     (F)

> for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations.

> 4) Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false [8]. They are exact solutions, not approximation. One counterexample is, of course, sufficient to disprove a conjecture.

> The following references include references used in the consolidated paper [8] plus [2] which applies [8]

> References

> [1]    Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
> [2]    Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
>         Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
> [3]    Corporate Mathematical Society of Japan , Kiyosi Itô, Encyclopedic dictionary of mathematics (2nd ed.), MIT Press, Cambridge, MA, 1993
> [4]    Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
> [5]    Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
> [6]    Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
> [7]    Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
> [8]    Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.  
> [9]    Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
>         Theory, Methods and Applications; online at Science Direct website
> [10]  Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
> [11]    Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
> [12]  Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis    (Elsevier Science, Ltd.), 2009, Paris.
> [13]  Counterexamples to Fermat’s last theorem,http://users.tpg.com.au/pidro/
> [14]  Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.

> E. E. Escultura
> Research Professor
> V. Lakshmikantham Institute for Advanced Studies
> GVP College of Engineering, JNT University
> Madurawada, Vishakhapatnam, AP, Indiahttp://users.tpg.com.au/pidro/