ARS - 2010
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Item Useful identities in finding a simple proof for Fermat’s last theorem(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Piyadasa, R.A.D.; Shadini, A.M.D.M.; Perera, B.B.U.P.Fermat’s last theorem, very famous and difficult theorem in mathematics, has been proved by Andrew Wiles and Taylor in 1995 after 358 years later the theorem was stated However, their proof is extremely difficult and lengthy. Possibility of finding s simple proof, first indicated by Fermat himself in a margin of his notes , has been still baffled and main objective of this paper is, however, to point out important identities which will certainly be useful to find a simple proof for the theorem.Item Effect of a long-ranged part of potential on elastic S-matrix element(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Shadini, A.M.D.M.; Piyadasa, R.A.D.; Munasinghe, J.It has been found that quantum mechanical three-body Schrödinger equation can be reduced to a set of coupled differential equations when the projectile can be easily breakable into two fragments when it is scattering on a heavy stable nucleus [1]. This coupled set of differential equations is solved under appropriate boundary conditions, and this method, called CDCC, has been found to be a very successful model in high energy quantum mechanical three body calculations [2]. It can be shown, however, that the coupling potentials in the coupled differential equations are actually long-range [3],[4] and asymptotic out going boundary condition, which is used to obtain elastic and breakup S-matrix elements is not mathematically justifiable. It has been found that the diagonal coupling potentials in this model takes the inverse square form at sufficiently large radial distances [3]and non-diagonal part of coupling potentials can be treated as sufficiently short-range to guarantee numeral calculations are feasible. Therefore one has to justify that the long range part of diagonal potential has a very small effect on elastic and breakup S-matrix elements to show that CDCC is mathematically sound .Although the CDCC method has been successful in many cases, recent numerical calculations[5],[6]indicate its unsatisfactory features as well. Therefore inclusion of the long range part in the calculation is also essential. The main objective of this contribution is to show that the effect of the long range part of the potentials on S-matrix elements is small.Item Simple proof of Fermat’s last theorem for n =11(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Shadini, A.M.D.M.; Piyadasa, R.A.D.Proof of Fermat’s last theorem for any odd prime is difficult. It may be extremely difficult to generalize any available Proof of Fermat’s last theorem for small prime such as n 3,5,7 to n 11[1]. The prime n 11 is different from n 13,17,19 in the sense that 2n 1 23 is also a prime and hence the corresponding Fermat equation may have only one type (Class.2) of solutions due to a theorem of Germaine Sophie[1],[2]. In this contribution, we will give a simple proof for the exponent n 11 based on elementary mathematics. The Darbrusow identity[1] that we will use in the proof can be obtained as Darbrusow did using the multinomial theorem on three components[1]. In our proof, it is assumed that the Fermat equation 11 11 11 z y x , (x, y) 1 has non-trivial integer solutions for (x, y, z) and the parametric solution of the equation is obtained using elementary mathematics. The proof of the theorem is done by showing that the necessary condition that must be satisfied by the parameters is never satisfied.Item A simple and short proof of Fermat’s last theorem for n = 3(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Shadini, A.M.D.M.; Piyadasa, R.A.D.The first proof of Fermat’s last theorem for the exponent n 3 was given by Leonard Euler. However, Euler did not establish in full the key lemma required in the proof [1]. Since then, several authors have published proof for the cubic exponent but Euler’s proof may have been supposed to be the simplest. Ribenboim [1] claims that he has patched up Euler’s proof and Edwards [2] also has given a proof of the critical and key lemma of Euler’s proof using the ring of complex numbers. Recently, Macys [3] in his article, claims that he may have reconstructed Euler are proof by providing an elementary proof for the key lemma. However, in this authors’ point of view, none of these proofs is short nor easy to understand compared to the simplicity of the wording and the meaning of the theorem.The main objective of this paper is, therefore, to provide a simple and short proof for the theorem. It is assumed that the equation , ( , ) 1 3 3 3 z y x x y has non-trivial integer solutions for (x, y, z) . Parametric solution of x, y, z and a necessary condition that must be satisfied by the parameters can be obtained using elementary mathematics. The necessary condition is obtained and the theorem is proved by showing that this necessary condition is never satisfied.