distribution of generalized fermat prime numbers numbers of the form fb n b2n 1 are called generalized fermat numbers gfn a computational method for testing the probable primality of a gfn is described which is as fast as testing a number of the form 2m 1 the theoretical distributions of gfn primes for fixed n are derived and compared to the actual distributions the predictions are surprisingly accurate and can be used to support bateman and horn s quantitative form of hypothesis h of schinzel and sierpinski a list of the current largest known gfn primes is included

riemannian theory of hamiltonian chaos and lyapunov exponents this paper deals with the problem of analytically computing the largest lyapunov exponent for many degrees of freedom hamiltonian systems this aim is succesfully reached within a theoretical framework that makes use of a geometrization of newtonian dynamics in the language of riemannian geometry a new point of view about the origin of chaos in these systems is obtained independently of homoclinic intersections chaos is here related to curvature fluctuations of the manifolds whose geodesics are natural motions and is described by means of jacobi equation for geodesic spread under general conditions ane effective stability equation is derived an analytic formula for the growth rate of its solutions is worked out and applied to the fermi pasta ulam beta model and to a chain of coupled rotators an excellent agreement is found the theoretical prediction and the values of the lyapunov exponent obtained by numerical simulations for both models

three new factors of fermat numbers we report the discovery of a new factor for each of the fermat numbers f 13 f 15 f 16 these new factors have 27 33 and 27 decimal digits respectively each factor was found by the elliptic curve method after division by the new factors and previously known factors the remaining cofactors are seen to be composite numbers with 2391 9808 and 19694 decimal digits respectively

a stochastic roundoff error analysis for the fast fourier transform we study the accuracy of the output of the fast fourier transform by estimating the expected value and the variance of the accompanying linear forms in terms of the expected value and variance of the relative roundoff errors for the elementary operations of addition and multiplication we compare the results with the corresponding ones for the direct algorithm for the discrete fourier transform and we give indications of the relative performances when different rounding schemes are used we also present the results of numerical experiments run to test the theoretical bounds and discuss their significance

large sophie germain primes if p is a prime and 2p 1 is also prime then p is a sophie germain prime in this article several new sophie germain primes are reported which are the largest known at this time the search method and the expected search times are discussed

exact enumeration approach to fractal properties of the percolation backbone and 1 σ expansion an exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold the authors use this approach to calculate exactly the first low density expansion of lbb p for arbitrary system dimensionality d where lbb p is the mean of backbone bonds and p is the bond occupation probability standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d these disagree with the sierpinski gasket model of the backbone they also calculate the first low density expansions of lmin p and lred p which are respectively the mean number of bonds in the minimum path between i and j and the mean number of singly connected red bonds

table errata 2 to factors of generalized fermat numbers we note that three factors are missing from table 1 in factors of generalized fermat numbers by a bjorn and h riesel published in math comp 67 1998 441 446

a new representation of elements of finite fields gf 2 sup m yielding small complexity arithmetic circuits let f sub 2 denote the binary field and f sub 2m an algebraic extension of degree m 1 over f sub 2 traditionally elements of f sub 2m are either represented as powers of a primitive element of f sub 2m together with 0 or by an expansion in a basis of the vector space f sub 2m over f sub 2 we propose a new representation based on an isomorphism from f sub 2m into the residue polynomial ring module x sup n 1 the new representation simultaneously satisfies the properties of various traditional representations which leads in some cases to architectures of parallel in parallel out arithmetic circuits adder multiplier exponentiator inverter squarer divider with average to small complexity we show that the implementation of all the arithmetic circuits designed for the new representation on an integrated circuit sometimes has smaller complexity than the implementation of all the arithmetic circuits designed for other representations in addition we derive a serial multiplier for the field f sub 2m which comprises the least number of gates of all the serial multipliers known to the author when m 1 is a prime such that 2 is primitive in the field z sub m 1

the fractal dimension of the minimum path in two and three dimensional percolation the authors calculate the fractal dimension d min of the shortest path l between two points on a percolation cluster where l approximately rd min and r is the pythagorean distance between the points they find d min 1 130 or 0 002 for d 2 and 1 34 or 0 01 for d 3

gradient method for thermal phase transitions the authors present a new numerical method of obtaining critical temperatures and exponents for ferromagnetic spin models and dynamical transitions this method is based on the spreading of damage in a temperature gradient they illustrate the method on the two dimensional ising model and present new results on the or ising spin glass in two dimensions they also obtain new critical exponents describing the behaviour of the width and the length of the damage front

parameter determination for complex number theoretic transforms using cyclotomic polynomials some new results for finding all convenient moduli m for a complex numbertheoretic transform with given transform length n and given primitive nth root of unity modulo m are presented the main result is based on the prime factorization for values of cyclotomic polynomials in the ring of gaussian integers

rapid multiplication modulo the sum and difference of highly composite numbers we extend the work of richard crandall et al to demonstrate how the discrete weighted transform dwt can be applied to speed up multiplication modulo any number of the form a b where p abp is small in particular this allows rapid computation modulo numbers of the form k 2n 1 in addition we prove tight bounds on the rounding errors which naturally occur in floating point implementations of fft and dwt multiplications this makes it possible for fft multiplications to be used in situations where correctness is essential for example in computer algebra packages

the twentieth fermat number is composite the twentieth fermat number f20 2220 1 has been proven composite by machine computation the fermat numbers are the numbers fn 224 1 orginally conjectured by fermat to be prime for all n in fact only for n equal to 0 through 4 are they known to be prime and small factors of f9 fl1 fl2 fl5 fl6 have been known for some time as part of a long term test of the hardware reliability of the cray 2 supercomputer at the supercomputing research center the authors proved that f20 2220 1 which had been the smallest fermat number of unknown character is composite the test for compositeness was the standard technique of pepin 5 for n 1 fn is prime if and only if 3 f 2 1 mod fn this test for compositeness does not of course produce factors of the number but was the test used for proving the compositeness of f7 f8 flo f13 f14 2 3 4 6 7 the result of the computation on the cray 2 has been verified by performing the same computation on a cray x mp belonging to cray research the total computation time on the cray 2 was about 10 cpu days the time on the cray xmp was 82 hours both programs ran as single processor programs on any available cpu of the respective machines the ability of either computer to run in parallel on multiple cpu s was not used the time needed to test fn for n in the range 10 through 20 is just slightly more than four times the time needed to test fn the number of multiplications doubles in incrementing n and the time required for each multiplication doubles being dependent almost entirely on the length of the operands our programs would thus determine the character of f22 which is now the smallest fermat number of unknown character in a little more than 16 times the time needed for our computation on f20 the table below summarizes what is now known about the fermat numbers for n less than or equal to 22 a status list for larger n appears in 1 this computation roughly one million squarings modulo a one million bit number would be impossible to do even on supercomputers without fast fourier transform techniques for integer multiplication since one reason for performing this computation was to verify hardware reliablity and not to minimize the execution time the program was written entirely in cray fortran and called cray library functions for the fft s the program itself was very simple and only about 200 lines long much of which was used for checkpointing and restarting the program the program was called into execution every time the cray 2 was restarted and so received april 6 1987 revised june 5 1987 1980 mathematics subject classification 1985 revnsion primary ilyii lla51 11 04 d1988 american mathematical society 0025 5718 88 1 00 25 per page

cumulant renormalisation group and its application to the incipient infinite cluster in percolation the authors introduce the cumulant method into the context of position space renormalisation group the advantage of this method over previous procedures is that one need not systematically evaluate all 2n states of each kadanoff cell where n is the number of elements sites or bonds in the cell they illustrate this method by calculating for bond percolation with an eight bond cell the critical exponents characterizing the following quantities i the mean number of backbone bonds ii the mean number of bonds in the minimum and maximum paths along the backbone iii the mean number of singly connected red bonds and iv the mean number of bonds in all the self avoiding walks connecting the two extreme points they derive previously obtained results of the conductivity and resistivity

the starry sky model of the percolation phase transition iii a new equilibrium equation and two kinds of the special elastic nodes abstract the equilibrium lame equation in two dimensional and three dimensional spaces can be presented as conductivity in magnetic field equation with lorenz force expression this allows us to apply the perturbation theory and generalize the lame equation and obtain a new equilibrium equation with all responses of compression and tension it means that compression in one direction leads to tension in the perpendicular direction and vice versa if the boundaries are fixed the solution in recurrent integral form for contributions to all orders of poisson ratio is then obtained zero order solution of lame equilibrium equation expresses the poisson ratio as ve p pc k k 0 3 and creates the special elastic nodes with maximum local deformation the first order solution gives mean values of effective lateral displacements or forces when lateral boundaries are fixed and expresses the bulk modulus as ke p pc 2t g where g 0 6 is a hall coefficient critical exponent it creates the special active elastic nodes where the whole sample microrotates around a new universality class of elasticity problem belongs to the hall coefficient universality class therefore the elasticity backbone coincides with the hall coefficient backbone with the fractal dimension d f b 2 25

area time efficient architecture of fft based montgomery multiplication the modular multiplication operation is the most time consuming operation for number theoretic cryptographic algorithms involving large integers such as rsa and diffie hellman implementations reveal that more than 75 percent of the time is spent in the modular multiplication function within the rsa for more than 1 024 bit moduli there are fast multiplier architectures to minimize the delay and increase the throughput using parallelism and pipelining however such designs are large in terms of area and low in efficiency in this paper we integrate the fast fourier transform fft method into the mclaughlin s framework and present an improved fft based montgomery modular multiplication mmm algorithm achieving high area time efficiency compared to the previous fft based designs we inhibit the zero padding operation by computing the modular multiplication steps directly using cyclic and nega cyclic convolutions thus we reduce the convolution length by half furthermore supported by the number theoretic weighted transform the fft algorithm is used to provide fast convolution computation we also introduce a general method for efficient parameter selection for the proposed algorithm architectures with single and double butterfly structures are designed obtaining low area latency solutions which we implemented on xilinx virtex 6 fpgas the results show that our work offers a better area latency efficiency compared to the state of the art fft based mmm architectures from and above 1 024 bit operand sizes we have obtained area latency efficiency improvements up to 50 9 percent for 1 024 bit 41 9 percent for 2 048 bit 37 8 percent for 4 096 bit and 103 2 percent for 7 680 bit operands furthermore the operating latency is also outperformed with high clock frequency for length 64 transform and above

evidence for non universal exponents in bootstrap percolation bootstrap percolation bp models are systems where sites are initially randomly occupied those sites that do not maintain a suitable local environment of at least m occupied sites are successively removed current open questions concerning these systems now focus on the universality of critical exponents for the lower m values simulations indicate that for m 3 on the simple cubic lattice the exponent beta which characterizes the critical behaviour of the percolation probability is distinct from that of usual random percolation m 0 the authors quote pc 0 5717 or 0 0005 and beta 0 6 or 0 1 for m 3 they argue that the connectivity exponent nu for m 2 should be the same as that of usual random percolation and within the limits of numerical accuracy the authors observe that this appears to be the case for m 3 on the simple cubic lattice the implications of this surprising dichotomy are considered

eden growth on fractal media abstract simulations are presented for two dimensional eden growth into fractal networks we find that the fractal dimension of the eden cluster d e does not have a simple behavior when the cluster grows into the structure of a fractal network we also find that d e depends strongly on the minimal path between two arbitrary points in the network backbone and on the number of paths that appear between these two points at each scale

generalised mersenne numbers revisited generalised mersenne numbers gmns were defined by solinas in 1999 and feature in the nist fips 186 2 and secg standards for use in elliptic curve cryptography their form is such that modular reduction is extremely efficient thus making them an attractive choice for modular multiplication implementation however the issue of residue multiplication efficiency seems to have been overlooked asymptotically using a cyclic rather than a linear convolution residue multiplication modulo a mersenne number is twice as fast as integer multiplication this property does not hold for prime gmns unless they are of mersenne s form in this work we exploit an alternative generalisation of mersenne numbers for which an analogue of the above property and hence the same efficiency ratio holds even at bitlengths for which schoolbook multiplication is optimal while also maintaining very efficient reduction moreover our proposed primes are abundant at any bitlength whereas gmns are extremely rare our multiplication and reduction algorithms can also be easily parallelised making our arithmetic particularly suitable for hardware implementation furthermore the field representation we propose also naturally protects against side channel attacks including timing attacks simple power analysis and differential power analysis which is essential in many cryptographic scenarios in constrast to gmns

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

factors of fermat numbers and large primes of the form 2ⁿ 1 a new factor is given for each of the fermat numbers f52 f931 f6835 and f9448 in addition a factor of f75 discovered by gary gostin is presented the current status for all f is shown in a table primes of the form k 2 1 k odd are listed for 31 k 149 1500 ii 4000 and for 151 k 199 1000 ii 4000 some primes for even larger values of ii are included the largest one being 5 213165 1 also a survey of several related questions is given in particular values of k such that k 2 i is composite for every ii are considered as well as odd values of h such that 3h 2 1 never yields a twin prime pair

the twenty second fermat number is composite we have shown by machine proof that f 22 2 222 1 is composite in addition we reenacted young and buell s 1988 resolution of f 20 as composite finding agreement with their final selfridge hurwitz residues we also resolved the character of all extant cofactors of f n n 22 finding no new primes and ruling out prime powers

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

new cullen primes numbers of the forms c n n 2 n 1 and w n n 2 n 1 are both called cullen numbers new primes c n are presented for n 4713 5795 6611 18496 for w n several new primes are listed the largest one having n 18885 furthermore all efforts made to factorize numbers c n and w n are described and the result the complete factorization for all n 300 is given in a supplement

factors of generalized fermat numbers generalized fermat numbers have the form f b m b 2m 1 their odd prime factors are of the form k 2 n 1 k odd n m it is shown that each prime is a factor of some f b m for approximately 1 k bases b independent of n divisors of generalized fermat numbers of base 6 base 10 and base 12 are tabulated three new factors of standard fermat numbers are included

a search for wieferich and wilson primes an odd prime p is called a wieferich prime if 2 p 1 1 mod p 2 alternatively a wilson prime if p 1 1 mod p 2 to date the only known wieferich primes are p 1093 and 3511 while the only known wilson primes are p 5 13 and 563 we report that there exist no new wieferich primes p 4 x 10 12 and no new wilson primes p 5x 10 8 it is elementary that both defining congruences above hold merely mod p and it is sometimes estimated on heuristic grounds that the probability that p is wieferich independently that p is wilson is about 1 p we provide some statistical data relevant to occurrences of small values of the pertinent fermat and wilson quotients mod p

speeding the pollard and elliptic curve methods of factorization since 1974 several algorithms have been developed that attempt to factor a large number n by doing extensive computations module n and occasionally taking gcds with n these began with pollard s p 1 and monte carlo methods more recently williams published a p 1 method and lenstra discovered an elliptic curve method ecm we present ways to speed all of these one improvement uses two tables during the second phases of p 1 and ecm looking for a match polynomial preconditioning lets us search a fixed table of size n with n 2 o n multiplications a parametrization of elliptic curves lets step 1 of ecm compute the x coordinate of np from that of p in about 9 3 1og2 n multiplications for arbitrary p

structure of parallel multipliers for a class of fields gf 2 m abstract this paper presents a configuration of parallel multipliers for gf 2 m based on canonical bases the possible parallel multipliers by the proposed configuration are limited to a class of fields gf 2 m however they can be constructed by o m 2 and gates and o m 2 eor gates with the structural modularity this is a desirable feature for the hardware implementation and their operation time is about log m t where m is the dimension of gf 2 m and t is the delay time of an eor gate in order to construct such parallel multipliers we define two types of polynomials of special form over gf 2 one is called all one polynomial denoted by aop and the other is called equally spaced polynomial denoted by esp furthermore we show a necessary and sufficient condition for esps to be irreducible over gf 2 and the uniqueness of the irreducible esps over gf 2 finally we propose the configuration of parallel multipliers for a class of fields gf 2 m based on irreducible aops and esps over gf 2

new frameworks for montgomery s modular multiplication method we present frameworks for fast modular multiplication based on a modification of montgomery s original method for fixed large integers our algorithms may be significantly faster than conventional methods our techniques may also be extended to modular polynomial arithmetic

factorization of the tenth fermat number we describe the complete factorization of the tenth fermat number f10 by the elliptic curve method ecm f10 is a product of four prime factors with 8 10 40 and 252 decimal digits the 40 digit factor was found after about 140 mflop years of computation we also discuss the complete factorization of other fermat numbers by ecm and summarize the factorizations

fast convolutions meet montgomery arithmetic in large ring and field extensions is an important problem of symbolic computation and it consists essentially of the combination of one multiplication and one division in the underlying ring methods are known for replacing one division by two short multiplications in the underlying ring which can be performed essentially by using convolutions however while using school book multiplication modular multiplication may be grouped into 2m r operations where m r denotes the number of operations of one multiplication in the underlying ring the short multiplication problem is an important obstruction to convolution it raises the costs in that case to 3m r in this paper we give a method for understanding and bypassing this problem thus reducing the costs of ring arithmetic to roughly 2m r when also using fast convolutions the algorithms have been implemented with results which fit well the theoretical prediction and which shall be presented in a separate paper

table errata new primality criteria and factorizations of 2 1 math comp 29 1975 620 647 mr 52 5546 by the author d h lehmer and j l selfridge

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

a new finite field multiplier using redundant representation this article presents simple and highly regular architectures for finite field multipliers using a redundant representation the basic idea is to embed a finite field into a cyclotomic ring which is based on the elegant multiplicative structure of a cyclic group one important feature of our architectures is that they provide area time trade offs which enable us to implement the multipliers in a partial parallel hybrid fashion this hybrid architecture has great significance in its vlsi implementation in very large fields the squaring operation using the redundant representation is simply a permutation of the coordinates it is shown that when there is an optimal normal basis the proposed bit serial and hybrid multiplier architectures have very low space complexity constant multiplication is also considered and is shown to have an advantage in using the redundant representation

factors of generalized fermat numbers a search for prime factors of the generalized fermat numbers f n a b a 2n b 2n has been carried out for all pairs a b with a b 12 and gcd a b 1 the search limit k on the factors which all have the form p k 2 m 1 was k 10 9 for m 100 and k 3 10 6 for 101 m 1000 many larger primes of this form have also been tried as factors of f n a b several thousand new factors were found which are given in our tables for the smaller of the numbers i e for n 15 or if a b 8 for n 16 the cofactors after removal of the factors found were subjected to primality tests an if composite with n 11 searched for larger factors by using the ecm and in some cases the mpqs ppmpqs or snfs as a result all numbers with n 7 are now completely factored

series expansion analysis of the backbone properties of two dimensional percolation clusters low density series expansions for the backbone properties of two dimensional bond percolation clusters are derived and analysed expansions for most of the 14 properties considered are new and are obtained to order on the square lattice and order on the triangular lattice earlier series work was confined to three properties of the square lattice and was to order the fractal dimension of the bonds or sites in the backbone is estimated to be and is intermediate between a previously conjectured field theory value and the latest monte carlo results the union intersection and length of the longest self avoiding paths are found to have the same fractal dimension which is close to and consistent with the field theory conjecture for on the other hand the union intersection and length of the shortest paths are found to have different dimensions and in the case of the intersection the triangular and square lattices are found to have sigificantly different dimensions the fractal dimension of the shortest path also appears to be non universal and we find for the square lattice and for the triangular lattice critical amplitude ratios are considered and found to be in agreement with theoretical inequalities

faster polynomial multiplication over finite fields polynomials over finite fields play a central role in algorithms for cryptography error correcting codes and computer algebra the complexity of multiplying such polynomials is still a major open problem let p be a prime and let mp n denote the bit complexity of multiplying two polynomials in fp x of degree less than n for n large compared to p we establish the bound mp n o n log n 8log n log p where log n min k n log k log n 1 stands for the iterated logarithm this improves on the previously best known bound mp n o n log n log log n log p which essentially goes back to the 1970s

effect of loops on the vibrational spectrum of percolation network abstract we study the effects of adding loops to a critical percolation cluster on the diffusional and equivalently scalar elastic properties of the fractal network from the numerical calculations of the eigenspectrum of the transition probability matrix we find that the spectral dimension ds and the walk dimension dw change suddenly as soon as the floppy ends of a critical percolation cluster are connected together to form relatively large loops and we find that the additional inclusion of successively smaller loops only change these exponents little if at all this suggests that there is a new universality class associated with the loop enhanced percolation problem

on the primality of n 1 and 2 3 5 p 1 for each prime p let p be the product of the primes less than or equal to p we have greatly extended the range for which the primality of n 1 and p 1 are known and have found two new primes of the first form 6380 1 6917 1 and one of the second 42209 1 we supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found

possibly exact fractal dimensions from conformal invariance based on an equivalence between the q state potts model and conformally invariant field theories the author speculates that all fractal dimensionalities of two dimensional percolation are given by the formula d 100 x2 48 where x is an integer the same formula should also apply to self avoiding walks and lattice animals this approach is extended to cluster weighted percolation his conjecture seems to agree with previous exact and numerical results except for the backbone exponent and possibly for lattice animals

distribution of dangling ends on the incipient percolation cluster we study numerically and by scaling arguments the probabilityp m dm that a given dangling end of the incipient percolation cluster has a mass between m and m dm we nd by scaling arguments that p m decays with a power law p m m 1 with an exponent d b f df where df and d b f are the fractal dimensions of the cluster and its backbone respectively our numerical results yield 0 83 in d 2 and 0 74 in d 3 in very good agreement with theory c 1999 published by elsevier science b v all rights reserved

regular random fractals and the n parameter model two models of fractal objects are presented the first model represents a new class of fractals which is intermediate to statistical fractals and exact fractals in that each configuration of the random ensemble has an exact and adjustable fractal dimension the second model has a continuously adjustable backbone dimensions and can be used to directly generate conducting backbones

the three dimensional ising model in a temperature gradient abstract the critical temperature t c and the correlation length exponent for the three dimensional ising model is determined via the gradient method by numerical studies of systems of size up to 360 3 we determine the correlation length exponent 0 62 0 01 in comparison to the best estimates ranging from 0 624 to 0 630

the twenty fourth fermat number is composite we have shown by machine proof that f24 2224 1 is composite the rigorous pepin primality test was performed using independently developed programs running simultaneously on two different physically separated processors each program employed a floating point fft based discrete weighted transform dwt to effect multiplication modulo f24 the final respective pepin residues obtained by these two machines were in complete agreement using intermediate residues stored periodically during one of the floating point runs a separate algorithm for pure integer negacyclic convolution verified the result in a wavefront paradigm by running simultaneously on numerous additional machines to effect piecewise verification of a saturating set of deterministic links for the pepin chain we deposited a final pepin residue for possible use by future investigators in the event that a proper factor of f24 should be discovered herein we report the more compact traditional selfridge hurwitz residues for the sake of completeness we also generated a pepin residue for f23 and via the suyama test determined that the known cofactor of this number is composite

divisors of mersenne numbers we add to the heuristic and empirical evidence for a conjecture of gillies about the distribution of the prime divisors of mersenne numbers we list some large prime divisors of mersenne numbers mp in the range 17000 p 105

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

shortest path fractal dimension for percolation in two and three dimensions we carry out a high precision monte carlo study of the shortest path fractal dimension d mathrm min for percolation in two and three dimensions using the leath alexandrowicz method which grows a cluster from an active seed site a variety of quantities are sampled as a function of the chemical distance including the number of activated sites a measure of the radius and the survival probability by finite size scaling we determine d mathrm min 1 130 phantom rule 0 16em 0ex 77 2 and 1 3756 6 in two and three dimensions respectively the result in two dimensions rules out the recently conjectured value d mathrm min 217 192 deng et al phys rev e 81 020102 r 2010

low weight polynomial form integers for efficient modular multiplication in 1999 solinas introduced families of moduli called the generalized mersenne numbers gmns which are expressed in low weight polynomial form p f t where t is limited to a power of 2 gmns are very useful in elliptic curve cryptosystems over prime fields since modular reduction by a gmn requires only integer additions and subtractions however since there are not many gmns and each gmn requires a dedicated implementation gmns are hardly useful for other cryptosystems here we modify gmn by removing restriction on the choice of t and restricting the coefficients of f t to 0 and plusmn1 we call such families of moduli low weight polynomial form integers lwpfis we show an efficient modular multiplication method using lwpfi moduli lwpfis allow general implementation and there exist many lwpfi moduli one may consider lwpfis as a trade off between general integers and gmns

irregular primes to one million using fast algorithms for power series inversion based on the fast fourier transform and multisectioning of power series we have calculated all irregular primes up to one million including their indices of irregularity and associated irregular pairs using this data we verified that fermat s last theorem and vandiver s conjecture are true for these primes two primes with index of irregularity five were already known we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six a pair of integers p k is said to be an irregular pair if p is a prime k is an even integer satisfying 2 p are known see 8 our calculations show that vandiver s conjecture is true for all p 106 the table of irregular pairs could also be used to calculate iwasawa invariants for the corresponding primes we did not do this as part of our calculations see 9 or 4 for a discussion of this problem previous computations of irregular pairs have used algorithms that take 0 p2 arithmetic operations for each prime p the bernoulli numbers are defined in received by the editor april 30 1991 and in revised form september 10 1991 1991 mathematics subject classification primary 11b68 secondary 1 1d41 65t20 1 1y99 the first author was partially supported by national science foundation grant dms 9012989 d 1992 american mathematical society 0025 5718 92 1 00 25 per page

order propagation near the percolation threshold a new monte carlo method for studying bond percolation clusters is developed and used to identify new critical quantities associated with the percolation threshold the bonds in each cluster are partitioned into three distinct connectivity classes red singly connected backbone bonds blue multiply connected backbone bonds and yellow non backbone bonds often called dangling ends among the new cluster properties studied are the mean number of red bonds a critical quantity diverging at p c with exponent y r approximately 1 and the length of the shortest connected path through the cluster which is critical with exponent y min 1 35 0 02 for all cluster properties studies the authors also compute averages over only the largest clusters the corresponding critical exponents are found to be significantly different from those obtained by averaging over clusters of all sizes

the k th prime is greater than k ln k ln ln k 1 for k 2 rosser and schoenfeld have used the fact that the first 3 500 000 zeros of the riemann zeta function lie on the critical line to give estimates on x and x with an improvement of the above result by brent et al we are able to improve these estimates and to show that the k th prime is greater than k ln k ln ln k 1 for k 2 we give further results without proof

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

field theoretic approaches to biconnectedness in percolating systems two field theoretic formulations for the percolation problem are presented from which the critical exponents describing the backbone of the infinite cluster at the percolation threshold are obtained at high spatial dimension d the order parameter exponent for the backbone beta 2 is given by beta 2 2 beta psi 2 nu where beta is the critical exponent for the density of the infinite cluster and psi 2 is a new crossover exponent in mean field theory psi 2 0 and for d 6 epsilon psi 2 2 epsilon 2 49 o epsilon 3 presumably psi 2 d is a smooth function of d for d d where numerical and theoretical work indicates that d is about 3 the result indicates that the fractal dimensionality of the backbone is given in terms of the percolation exponents as gamma nu psi 2

time dependent thermodynamic properties of the ising model from damage spreading the relationship between damage spreading and static thermodynamic properties in the ising model developed by coniglioet al is here extended to include time dependent thermodynamic quantities we exploit this new result to measure the time dependent spin correlation function from damage spreading in the ising model with heat bath and glauber dynamics until now only static thermodynamic quantities have been correctly determined from damage spreading and even then only with heat bath dynamics we also show that there are significant differences between the kinetics of damage spreading as found in heat bath and glauber dynamics

new fibonacci and lucas primes extending previous searches for prime fibonacci and lucas numbers all probable prime fibonacci numbers f n have been determined for 6000 1000 as far as they have been completed adding information to existing factor tables covering n 1000

geometric approach to chaos in the classical dynamics of abelian lattice gauge theory a riemannian geometrization of dynamics is used to study chaoticity in the classical hamiltonian dynamics of a u 1 lattice gauge theory this approach allows one to obtain analytical estimates of the largest lyapunov exponent in terms of time averages of geometric quantities these estimates are compared with the results of numerical simulations and turn out to be very close to the values extrapolated for very large lattice sizes even when the geometric quantities are computed using small lattices the scaling of the lyapunov exponent with the energy density is found to be well described by the law