ARS - 2011
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Item Integer roots of two polynomial equations and a simple proof of Fermat‟s last theorem(University of Kelaniya, 2011) Piyadasa, R.A.D.; Perera, B.B.U.P.Fermat‟s last theorem (FLT),possibly written in 1637,despite its rather simple statement, is very difficult to prove for general exponent n [1]. In fact, formal complete proof of FLT remained illusive until 1995 when Andrew Wiles and Taylor[1],[2] put forward one based on elliptic curves[3]. It is well known that their proof is lengthy and difficult to understand. Main objective of this paper is to provide a simpler and shorter proof for FLT. It is shown that FLT can be proved by showing that two polynomial equations have no integer roots when the independent variable satisfies certain conditions. Theorem: The polynomial equations in x 2. ( ) 0 1 p m p pm p x p uhdx h p u 2 ( ) 0 p m p p x uhdp x h u where u,h, p,d are integers co-prime to one another , p is an odd prime and m 2 , have no integer roots co-prime to h for any integer values of it when u,h are both odd or of opposite parity[4],[5]. Lemma If ( , ) 0(mod ) p p m F a b a b p and (a, p) (b, p) 1, then 0(mod ) 1 m a b p and m 2 Proof of the theorem: We first consider the equation 2. ( ) 0 1 p m p pm p x p uhdx h p u The integer roots of this equation are the integer factors of p pm p h p u 1 and let us assume that it has an integer root. This integer root obviously must be co-prime to u,h, p since they are co-prime. If an integer satisfies the equation , then ( ) 2. ) 0 1 p p m pm p g h p uhd p u and 0(mod ) 1 m g h p . Therefore, we can write g h p j m1 , where the integer j is co-prime to d,h, p .Now, our equation can be written as m m p m p pm p h p j h p j uhdp h p u 1 1 1 1 ( )[( ) 2 ] and we use the remainder theorem to check weather the linear factor h p j m1 in h a factor of the polynomial p pm p h p u 1 in h . If so, 0 1 pm p p pm p p j p u This is impossible since ( j, p) 1, and we conclude that (1) has no integer roots we need. If g satisfies the equation 2 ( ) 0 p m p p g uhdp g h u and g must be a factor of p p h u . We also assume that h,u are both odd or of opposite parity which is relevant to Fermat‟s last theorem. First of all, we will show that g h u using the relation i i p i p i i p p p i p i C u h h u i p h u h u 2 2 1 1 1 1 ( ) .( 1) . ( ) Our equation takes the form g h u ughdp If g h u 0 , then we must have ( ) - (-1) ] 0 2 - 2 - [ .( ) 2 -3 2 -3 2 -1 -5 1 -3 -3 p p p m p p p C uh h u p u h p dp p h u If both u and h are odd, or , of opposite parity ,then the term ( ) - (-1) ] 2 -[ .( ) 2 -3 2 -3 2 -1 -4 1 -3 -3 p p p p p p C uh h u p u h p p h u is odd since p( 3) is an odd prime and therefore the equation ( ) - (-1) ] 0 2 - 2 - [ .( ) 2 -3 2 -3 2 -1 -5 1 -3 -3 p p p m p p p C uh h u p u h p dp p h u will never be satisfied since m 2dp is even. Hence, g h u . From the equation ( ) 2 .( 1) . . ( ) 0 2 1 1 2 1 1 i i p i i i p i p i p p m C u h h u i p g h u ughdp we conclude that g (h u) 0(mod p) , which follows from the lemma Therefore , we can write g h (u p j) k , where k 1 , j 0 and ((u p j),h) 1 k is since (g,h) 1. Since g h (u p j) k is an integer root of the equation we can write [ ( )][( ) 2 ] p p k k p 1 m h u h u p j h u p j uhdp As before , using the remainder theorem, we get ( ) 0 k p p u p j u But this equation will never be satisfied since j 0 .Item Possible quark confinement by a non-relativistic model(University of Kelaniya, 2011) Karunatathne, S.; Piyadasa, R.A.D.Confinement of quarks by an infinitely deep potential well is well [1] known. We are interested in confinement of quarks by the singular potential 2 1 r when the effective potential , 2 2 ( 1) r r l l is negative, where 2 1 2 . However, we have found that the corresponding series solution is not a bound wave function. Now, we assume that quarks are localized to a small region and obtain the bound states in the following way. Consider the Schrödinger equation in the form 0 ( 1) 2 2 2 2 2 u r r l l k dr d u (1.1) and let us choose such l(l 1) . Then (1.1) reduces to 0 2 2 2 k u dr d u and the wave function kr u r e ( ) and the total radial wave function is given by R(r) = r e r u r kr ( ) (1.2) which is normalizable and the normalization constant 2 1 (2k) . We conclude that non relativistic quarks having nonzero angular momentum can be bound by the inverse square potential and the quark wave function can be made highly localized acquiring sufficient energy 2 2 2 k . We use the experimental value of the size(diameter) of the nucleon of 1.6 fm to determine the value of k . We can attribute this value to the mean square radius given by 2 | | 0.64 2 2 2 0 2 r kr e dr r kr (1.3) The equation (1.3) gives 2 (0.64) 1 2 k (1.4) We have assumed that the quark mass is and therefore the quark binding energy E is given by 2 2 2 k , and if we use to be one third of the nucleon mass , then E 62 48.437 2 938 197. 197 3 2 2 0.556 fm 2k 1 r Strength of the potential can be found in this case by using l(l 1) . If l 1, 2 2 2 1 . Therefore 60MeV 938 . 3.200.200 2 1 Another important point to be mentioned here is that attractive potential can be bound to potential centre of a circular orbit by an inverse square potential only if the total energy of the particle is zero in case of classical mechanics. Therefore, our quark bound states might be stable if they are confined to a very small region and they are undisturbed. This conclusion is actually based on classical mechanics but plausible since speeds of quarks should be big.Item Solution of a special Diophantine equation using elementary mathematics(12th Annual Research Symposium, University of Kelaniya, 2011) Piyadasa, R.A.D.Item Uniqueness of roots of a cubic and proof of Fermat‟s last theorem for n=3(12th Annual Research Symposium, University of Kelaniya, 2011) Shadini, A.M.D.M.; Piyadasa, R.A.D.; Perera, B.B.U.P.Item The inverse square potential and relativistic bound states(12th Annual Research Symposium, University of Kelaniya, 2011) Karunarathne, Sanjeewa; Piyadasa, R.A.D.Item Simple proof of two important theorems in number theory(12th Annual Research Symposium, University of Kelaniya, 2011) Jayani; Manike, K.R.C.; Piyadasa, R.A.D.