ARS - 2010
Permanent URI for this collectionhttp://repository.kln.ac.lk/handle/123456789/168
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Item A simple analytical proof of Fermat’s last theorem for n = 7(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Piyadasa, R.A.D.It is well known that proof of Fermat’s last theorem for any odd prime is difficult and first proof for n 7 was given by Lame [1] ,and Kumar also has given a proof for a special class of primes (Regular primes)which includes the case n 7 .However, these proofs are lengthy and difficult and may not easily be extended for all odd primes. The prime n 7 differs from n 5 since 2.7.115 is not a prime, whereas 2.5111 is a prime. Then it follows from the famous theorem of Germain Sophie that the corresponding Fermat’s equation , ( , ) 1 7 7 7 z x y x y may have two classes of integer solutions, xyz 0(mod7) and xyz 0(mod7) if we assume that the Fermat equation has non-trivial integer solutions for x, y, z . This fact is proved using the simple argument [3] of Oosterhuis. The main objective of this paper is to give a simple analytical proof for the Fermat’s last theorem n 7 using general respective parametric solutions corresponding two classes of solutions of the Fermat equation ,which has already been extended for all odd primes.Item Method of Infinite Descent and proof of Fermat's last theorem for n = 3(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Piyadasa, R.A.D.The first proof of Fermat’s last theorem for the exponent n 3 was given by Leonard Euler using the famous mathematical tool of Fermat called the method of infinite decent. However, Euler did not establish in full the key lemma required in the proof. Since then, several authors have published proofs for the cubic exponent but Euler's proof may have been supposed to be the simplest. Paulo 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 in his recent article [3, Eng.Transl.] claims that he may have reconstructed Euler’s proof for the key lemma. However, none of these proofs is short nor easy to understand compared to the simplicity of the theorem and the method of infinite decent The main objective of this paper is to provide a simple, short and independent proof for the theorem using the method of infinite decent. It is assumed that the equation 3 3 3 z y x , (x, y) 1 has non trivial integer solutions for (x, y, z) and their parametric representation [5] is obtained with one necessary condition that must be satisfied by the parameters. Using this necessary condition, an analytical proof of the theorem is given using the method contradiction. The proof is based on the method of finding roots of a cubic formulated by Tartagalia and Cardan [4], which is very much older than Fermat’s last theorem.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.Item The Equality of Schrödinger’s Theory and Heisenberg’s S-matrix Theory(Research Symposium 2010 - Faculty of Graduate Studies, University of Kelaniya, 2010) Silva, H. I. R. U.; Piyadasa, R.A.D.It is well known that the Schrödinger’s equation can be solved in few cases of physical importance [1] . Nevertheless, S-matrix theory can be used in general to describe physically important variables such as differential cross section, total cross section, etc….[2]. Since there’s no any justification of theoretical work to the best of our knowledge to verify that the Schrödinger’s theory and Heisenberg’s S-matrix theory are equivalent in case of important interacting potentials for which the Schrödinger’s equation can be solved analytically, we have used Heisenberg’s S-matrix theory and Schrödinger’s wave mechanics to justify that the two theories give exactly the same eigenvalues in cases which we have examined. To obtain them, we were able to find the discrete energy eigenvalues in closed form in Heisenberg’s theory without graphical methods.