a new criterion for the first case of fermat s last theorem a new criterion of the first case of fermat last theorem is derived this criterion is based on special sums of reciprocals

on the number of false witnesses for a composite number if u is not a multiple of n and a l 1 mod n 1 then n must be composite and u is called a witness for n let f n denote the number of false witnesses for n that is the number of u mod n with ci 1 mod n considered here is the normal and average size of f n for i composite also considered is the situation for the more stringent euler and strong pseudoprime tests

analysis of semi implicit dgfem for nonlinear convection diffusion problems on nonconforming meshes the paper deals with the numerical analysis of a scalar nonstationary nonlinear convection diffusion equation the space discretization is carried out by the discontinuous galerkin finite element method dgfem on general nonconforming meshes formed by possibly nonconvex elements with nonsymmetric treatment of stabilization terms and interior and boundary penalty the time discretization is carried out by a semi implicit euler scheme in which the diffusion and stabilization terms are treated implicitly whereas the nonlinear convective terms are treated explicitly we derive a priori asymptotic error estimates in the discrete l l2 l l2 norm l2 h1 l2 h1 seminorm and l h1 l h1 seminorm with respect to the mesh size h and time step numerical examples demonstrate the accuracy of the method and manifest the effect of nonconvexity of elements and nonconformity of the mesh

imex hdg dg a coupled implicit hybridized discontinuous galerkin and explicit discontinuous galerkin approach for shallow water systems abstract we propose imex hdg dg schemes for planar and spherical shallow water systems of interest is subcritical flow where the speed of the gravity wave is faster than that of nonlinear advection in order to simulate these flows efficiently we split the governing system into a stiff part describing the gravity wave and a non stiff part associated with nonlinear advection the former is discretized implicitly with the hdg method while an explicit runge kutta dg discretization is employed for the latter the proposed imex hdg dg framework 1 facilitates high order solutions both in time and space 2 avoids overly small time step sizes 3 requires only one linear system solve per time stage 4 relative to dg generates smaller and sparser linear systems while promoting further parallelism and 5 suppresses the fast modes in the system with a large time step size numerical results for various test cases demonstrate that our methods are beneficial for applications where non stiff terms are accurately treated while stiff terms are less accurately handled

a discontinuous hp finite element method for diffusion problems 1 d analysis abstract this paper presents the mathematical analysis of a new variant of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems not requiring auxiliary variables such as those used in mixed methods the focus of this study is on a class of linear second order boundary value problems for which we prove stability and a priori error estimates in both the finite and infinite dimensional spaces

on fermat s quotient base two this paper extends the search for solutions of the congruence 2p1 0 modp2 to the limit p 6 109 no solution except the well known p 1093 and p 3511 was found in 1969 brillhart tonascia and weinberger 1 reported on the search for solutions of the congruence a 1 modp2 a 2 1 199 for a 2 a special effort was made to consider all p 3 109 only the two known solutions p 1093 and p 3511 were found having the occasional opportunity to use the illiac iv one of the projects decided upon was the recalculation and extension of the above result for a 2 by a somewhat different method the calculation was pushed to twice the above limit that is top 6 109 without finding any further solutions the parallel construction of the illiac iv makes it possible to look for 64 different values of p at the same time the speed of illiac iv is such that a range of 100000 numbers can be searched in one second thus the range for p 6 109 was broken up into 60 runs of 1000 seconds each the program was run as one of a few backlog problems over the past two years let n belong to one of the 64 residue classes that are prime to 240 and let 2m am nbm mod n2 where 0 am n 0 bm n for numbers n as large as 109 the number n2 is a doubly precise integer nevertheless the calculation of am and bm can be accomplished by single precision arithmetic in only o log m operations in fact one uses one or the other of the following two recurrences if m 2h 1 then am 2a2h and bm 2b2h mod n if m 2h and if a 2 rm nqm 0 rm n and if 2ahbh dm mod n then am rm and bm qm dm received march 30 1980 1980 mathematics subject classification primary 10 04 1981 american mathematical society 0025 5718 81 0000 0027 0 1 50 289 this content downloaded from 157 55 39 58 on tue 11 oct 2016 04 26 58 utc all use subject to http about jstor org terms

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

a staggered semi implicit discontinuous galerkin method for the two dimensional incompressible navier stokes equations in this paper we propose a new spatially high order accurate semi implicit discontinuous galerkin dg method for the solution of the two dimensional incompressible navier stokes equations on staggered unstructured curved meshes while the discrete pressure is defined on the primal grid the discrete velocity vector field is defined on an edge based dual grid the flexibility of high order dg methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non zero blocks per element for only one scalar unknown namely the pressure the method is computationally efficient since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions furthermore all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage which leads to savings in terms of computational effort for the time evolution part in this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step the method is validated for polynomial degrees up to p 3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data

analysis of a bdf dgfe scheme for nonlinear convection diffusion problems we deal with the numerical solution of a scalar nonstationary nonlinear convection diffusion equation we employ a combination of the discontinuous galerkin finite element method for the space semi discretization and the k step backward difference formula for the time discretization the diffusive and stabilization terms are treated implicitly whereas the nonlinear convective term is treated by a higher order explicit extrapolation method which leads to the necessity to solve only a linear algebraic problem at each time step we analyse this scheme and derive a priori asymptotic error estimates in the discrete l l 2 norm and the l 2 h 1 seminorm with respect to the mesh size h and time step for k 2 3 numerical examples verifying the theoretical results are presented

on euclidean ideal classes in certain abelian extensions in this article we show that certain abelian extensions k with unit rank greater than or equal to three have cyclic class group if and only if it has a euclidean ideal class this result improves an earlier result of murty and graves one can improve this result up to unit rank 2 if one assumes the elliott and halberstam conjecture see conjecture 1 in preliminaries these results are known under generalized riemann hypothesis by the work of lenstra j lond math soc 10 457 465 see also weinberger proc symp pure math 24 321 332

a special extension of wieferich s criterion the following theorem is proved in this paper if the first case of fermat s last theorem does not hold for sufficiently large prime 1 then e x1 2 kl x k x i 0 modl1 x nn for all pairs of positive integers n k n 94 0 k ni the proof of this theorem is based on a recent paper of skula and uses computer techniques 0 introduction the first case of fermat s last theorem states that for each odd prime i the equation xi y1 zi 0 has no integral solution x y z with i t xyz one of many methods investigating this problem was introduced by a wieferich this method is connected with the fermat quotients ql a al l q a i defined for each integer a such that a is not divisible by 1 let us assume in this paragraph that is an odd prime which does not satisfy the first case of fermat s last theorem in 1909 wieferich 7 published the following important result q 2 0 mod1 many mathematicians have extended this wieferich criterion the latest result is due to a granville and b monagan 1 and states q1 p 0 mod 1 for each prime p such that p 89 these considerations have been generalized by l skula he studied the sums s k n e 1 2 kl x k 1 1 s k n ex x

new bound for the first case of fermat s last theorem we present an improvement to gunderson s function which gives a lower bound for the exponent in a possible counterexample to the first case of fermat s last theorem assuming that the generalized wieferich criterion is valid for the first n prime bases the new function increases beyond n 29 unlike gunderson s and it increases more swiftly using the recent extension of the wieferich criterion to n 24 by granville and monagan the first case of fermat s last theorem is proved for all prime exponents below 156 442 236 847 241 729

discontinuous galerkin method for the solution of a transport level set problem the subject of the paper is the numerical analysis of the transport level set problem discretized by the discontinuous galerkin method without the assumption that the first order nonstationary transport equation contains a reaction term which is used in a standard literature we prove error estimates in the l l 2 norm in the case of the space semidiscretization method of lines and in the case of the space time discontinuous galerkin method in the l 2 l 2 norm numerical experiments support the derived error estimates and show that they are not sharp in the case of the space time discontinuous galerkin method

error estimates for a combined finite volume finite element method for nonlinear convection diffusion problems the subject of this paper is the analysis of error estimates of the combined finite volume finite element fv fe method for the numerical solution of a scalar nonlinear conservation law equation with a diffusion term nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume mesh dual to a triangular grid whereas the diffusion term is discretized by piecewise linear conforming triangular finite elements under the assumption that the exact solution possesses some regularity properties and the triangulations are of a weakly acute type with the aid of the discrete maximum principle and a priori estimates error estimates of the method are proved

on the method of coleman and chabauty let c be a curve of genus g 2 defined over a number field k and let j be the jacobian of c coleman c2 following chabauty has shown how to obtain good bounds on the cardinality of c k if the rank r of the mordell weil group j k is less than g the key to the method is to construct a logarithm on j kv for some valuation v of k whose kernel contains j k and whose restriction to c kv is represented explicitly as the integral of a differential this paper is an attempt to make the case through a detailed examination of the case of fermat curves that this method can be fashioned into a quite precise tool for bounding rational points on curves we show how to transform an element of the selmer group of the jacobian of a fermat curve of degree 19 into a p adic analytic function on the curve itself whose zero set contains all the rational points as a consequence we prove the second case of fermat s last theorem for regular primes the method depends on the existence of a suitable element in the selmer group for the lack of a satisfactory theory of descent for jacobians of fermat curves we can only show that this element exists in the case that p is regular of course in that case kummer had already proved the whole of fermat s last theorem however we believe the interest of this paper is in the method not the theorem and as such is independent of kumaner and also of the recent work of wiles our method is different and offers a development of the method of coleman and chabauty many aspects of which are generalizable to arbitrary curves although we do not attempt to make that generalization here we now describe the contents of the paper in more detail let p be an odd prime and let f be the pm fermat curve with projective equation

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows i general formulation a new space time discontinuous galerkin finite element method for the solution of the euler equations of gas dynamics in time dependent flow domains is presented the discontinuous galerkin discretization results in an efficient elementwise conservative upwind finite element method which is particularly well suited for local mesh refinement the upwind scheme uses a formulation of the hllc flux applicable to moving meshes and several formulations for the stabilization operator to ensure that monotone solutions around discontinuities are investigated the non linear equations of the space time discretization are solved using a multigrid accelerated pseudo time integration technique with an optimized runge kutta method the linear stability of the pseudo time integration method is investigated for the linear advection equation the numerical scheme is demonstrated with simulations of the flow field in a shock tube a channel with a bump and an oscillating naca 0012 airfoil these simulations show that using the data at the superconvergence points the accuracy of the numerical discretization is o h5 2 in space for smooth subsonic flows both on structured and on locally refined meshes and that the space time adaptation can significantly improve the accuracy and efficiency of the numerical method

semi implicit discontinuous galerkin methods for the incompressible navier stokes equations on adaptive staggered cartesian grids in this paper a new high order semi implicit discontinuous galerkin method si dg is presented for the solution of the incompressible navier stokes equations on staggered space time adaptive cartesian grids amr in two and three space dimensions the pressure is written in the form of piecewise polynomials on the main grid which is dynamically adapted within a cell by cell amr framework according to the time dependent main grid different face based spatially staggered dual grids are defined for the piece wise polynomials of the respective velocity components although the resulting adaptive staggered grids are more complex than classical uniform cartesian meshes the numerical scheme can still be written in a rather compact form thanks to the use of a tensor product formulation for the definition of the nodal basis in the d dimensional space d 2 3 all the discrete operators can be efficiently written as a combination of linear one dimensional operators acting in the d space directions separately arbitrary high order of accuracy is achieved in space while a very simple semi implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation the real advantages of the staggered grid arise in the solution of the schur complement associated with the saddle point problem of the discretized incompressible navier stokes equations i e after substituting the discrete momentum equations into the discrete continuity equation this leads to a linear system for only one unknown the scalar pressure indeed the resulting linear pressure system is shown to be symmetric and positive definite in order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor an implicit discretization is also used for the diffusive terms in the momentum equation a particular feature of our staggered dg approach is that the viscous stress tensor is discretized on the dual mesh this corresponds to the use of a lifting operator but on the staggered grid both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix free conjugate gradient method for which fast convergence is observed moreover it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner due to the explicit discretization of the nonlinear convective terms the final algorithm is stable within the classical cfl type time step restriction for explicit dg methods based on the fluid velocity moreover by using a suitable local time stepping technique based on the ader dg framework each element can use its optimal local time step also within the amr context the new space time adaptive staggered dg scheme has been thoroughly verified for polynomial degrees up to n 9 for a large set of non trivial test problems in two and three space dimensions for which analytical numerical or experimental reference solutions exist the high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution to the knowledge of the authors this is the first staggered semi implicit dg scheme for the incompressible navier stokes equations on space time adaptive meshes in two and three space dimensions

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

analysis of the discontinuous galerkin method for nonlinear convection diffusion problems abstract the subject matter is the analysis of the discontinuous galerkin finite element method applied to a nonlinear convection diffusion problem in the contrary to the standard fem the requirement of the conforming properties is omitted this allows us to consider general polyhedral elements with mutually disjoint interiors we do not require their convexity but assume only that they are star shaped we present an error analysis for the case of a nonsymmetric discretization of diffusion terms theoretical results are accompanied by numerical experiments

b expansions and symbolic dynamics given a real number p i it is easy to expand x e o i in a canonical way into a series of negative powers of p with integer coefficients less than or equal to p the form of the set of such p expansions which is a matter of symbolic dynamics or language theory varies according to arithmetical properties of p but there are many common features known facts which are few and open questions much more numerous are reviewed in this paper

a staggered semi implicit spectral discontinuous galerkin scheme for the shallow water equations a spatially arbitrary high order semi implicit spectral discontinuous galerkin dg scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed the free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis and as test functions each unknown is however defined on a different set of control volumes that are spatially staggered with respect to each other a semi implicit time integration yields a stable and efficient mass conservative algorithm the use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block penta diagonal system must be solved for the new free surface location which is a scalar quantity subsequently the new momentum components are directly obtained the staggered semi implicit approach makes the present dg scheme different from other published dg schemes the sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix free gmres algorithm furthermore for the shallow water equations the proposed scheme can be written as a quadrature free method where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors for the special case n 0 the proposed method reduces to a classical semi implicit finite difference scheme the proposed semi implicit scheme is particularly well suited for low froude number flows the method is validated on some typical academic benchmark problems using polynomial degrees of up to n 20

analysis of space time discontinuous galerkin method for nonlinear convection diffusion problems the paper presents the theory of the discontinuous galerkin finite element method for the space time discretization of a nonstationary convection diffusion initial boundary value problem with nonlinear convection and linear diffusion the problem is not singularly perturbed with dominating convection the discontinuous galerkin method is applied separately in space and time using in general different space grids on different time levels and different polynomial degrees p and q in space and time dicretization in the space discretization the nonsymmetric symmetric and incomplete interior and boundary penalty nipg sipg iipg approximation of diffusion terms is used the paper is concerned with the proof of error estimates in l 2 l 2 and dg norm formed by the l 2 h 1 seminorm and penalty terms a special technique based on the use of the gauss radau interpolation and numerical integration has been used for the derivation of an abstract error estimate in the dg norm the error estimates are optimal with respect to the size of the space grid they are optimal with respect to the time step if the dirichlet boundary condition has behaviour in time as a polynomial of degree q

the first case of fermat s last theorem is true for all prime exponents up to 714 591 416 091 389 we show that if the first case of fermat s last theorem is false for prime exponent p then p2 divides qp q for all primes q sq as a corollary we state the theorem of the title 1988 american mathematical society

a high order semi implicit discontinuous galerkin method for the two dimensional shallow water equations on staggered unstructured meshes a well balanced spatially arbitrary high order accurate semi implicit discontinuous galerkin scheme is presented for the numerical solution of the two dimensional shallow water equations on staggered unstructured non orthogonal grids the semi implicit method is derived in such a fashion that all relevant integrals can be precomputed and stored in a preprocessing stage so that the extension to curved isoparametric elements is natural and does not increase the computational effort of the simulation at runtime for p 0 the resulting scheme becomes a generalization of the classical semi implicit finite volume finite difference scheme of casulli and walters 2000 25 but with less conditions on the grid geometry the method proposed in this paper allows large time steps with respect to the surface wave speed gh and is thus particularly suitable for low froude number flows the approach is validated on some typical academic benchmark problems using polynomial degrees up to p 6

a class of implicit explicit two step runge kutta methods this work develops new implicit explicit time integrators based on two step runge kutta methods the class of schemes of interest is characterized by linear invariant preservation and high stage orders theoretical consistency stability and stiff convergence analyses are performed to reveal the excellent properties of these methods the new framework offers extreme flexibility in the construction of partitioned integrators since no coupling conditions are necessary practical schemes of orders three four and six are constructed and are used to solve several test problems numerical results confirm the theoretical findings

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

on the factors of the sturmian sequences resume cet article a pour objet l etude d une construction associant a toute droite de pente p q p et q premiers entre eux et q n un mot de longueur n sur l alphabet 0 1 nous montrons que nous obtenons par cette construction le langage constitue de tous les facteurs des suites de sturm nous formulons apres avoir obtenu une equation fonctionelle dont la solution est la serie generatrice de ce langage une conjecture reliant cette serie generatrice a la fonction d euler

gunderson s function in fermat s last theorem we study gunderson s function which gives a bound on the first case of fermat s last theorem assuming that the generalized wieferich criterion is valid for the first n prime bases we note two unexpected phenomena

language complexity of rotations and sturmian sequences given a rotation of the circle we study the complexity of formal languages that are generated by the itineraries of interval covers these languages are regular iff the rotation is rational in the case of irrational rotations our study reduces to that of the language complexity of the corresponding sturmian sequences we show that for a large class of irrationals including e all quadratic numbers and more generally all hurwitz numbers the corresponding languages can be recognized by a nondeterministic turing machine in linear time in other words belongs to nlin

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments