the rabin monier theorem for lucas pseudoprimes we give bounds on the number of pairs p q with 0 p q n such that a composite number n is a strong lucas pseudoprime with respect to the parameters p q

notes on some new kinds of pseudoprimes j browkin defined in his recent paper math comp 73 2004 pp 1031 1037 some new kinds of pseudoprimes called sylow p pseudoprimes and elementary abelian p pseudoprimes he gave examples of strong pseudoprimes to many bases which are not sylow p pseudoprime to two bases only where p 2 or 3 in this paper in contrast to browkin s examples we give facts and examples which are unfavorable for browkin s observation to detect compositeness of odd composite numbers in section 2 we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also sylow 2 pseudoprimes to the same bases in section 3 we give examples of sylow p pseudoprimes to the first several prime bases for the first several primes p we especially give an example of a strong pseudoprime to the first six prime bases which is a sylow p pseudoprime to the same bases for all p 2 3 5 7 11 13 in section 4 we define n to be a k fold carmichael sylow pseudoprime if it is a sylow p pseudoprime to all bases prime to n for all the first k smallest odd prime factors p of n 1 we find and tabulate all three 3 fold carmichael sylow pseudoprimes 10 16 in section 5 we define a positive odd composite n to be a sylow uniform pseudoprime to bases b 1 b k or a syl upsp b 1 b k for short if it is a syl p psp b 1 b k for all the first w n 1 1 small prime factors p of n 1 where w n 1 is the number of distinct prime factors of n 1 we find and tabulate all the 17 syl upsp 2 3 5 s 10 16 and some syl upsp 2 3 5 7 11 s 10 24 comparisons of effectiveness of browkin s observation with miller tests to detect compositeness of odd composite numbers are given in section 6

rabin miller primality test composite numbers which pass it the rabin miller primality test is a probabilistic test which can be found in several algebraic computing systems such as pari maple scratch pad because it is very easy to implement and with a reasonable amount of computing indicates whether a number is composite or probably prime with a very low probability of error in this paper we compute composite numbers which are strong pseudoprimes to several chosen bases because these bases are those used by the scratchpad implementation of the test we obtain by a method which differs from a recent one by jaeschke composite numbers which are found to be probably prime by this test 1 preliminaries

exact and coexact matrices 1 f r r is a c map here k is the real numbers r and r is the ring of c maps from r to r 2 f c c is an analytic map here k is the complex numbers c and r is the ring of analytic maps from c to c 3 f k k is a polynomial map here k is a commutative ring and r is the polynomial ring kc 4 f k i k nll is a k algebra homomorphism here k is a commutative ring and r is the formal power series ring kccnl

the combinatorics of coxeter categories we present an alternative construction of soergel s category of bimodules associated to a reflection faithful representation of a coxeter system we show that its objects can be viewed as sheaves on the associated moment graph we introduce an exact structure and show that soergel s special bimodules are the projective objects then we construct the indecomposable projectives by both a global and a local method discuss a version of the kazhdan lusztig conjecture and prove it for universal coxeter systems

tangent cones to schubert varieties in types an bn and cn abstract we study tangent cones to schubert subvarieties of the flag variety of a complex reductive group g let t be a maximal torus of g b be a borel subgroup of g containing t be the root system of g with respect to t w be the weyl group of and f g b be the flag variety we prove that if every irreducible component of is of type b n or c n and w 1 w 2 are two distinct involutions in w then the tangent cones at the point p e b to the corresponding schubert subvarieties x w 1 x w 2 of f do not coincide as subschemes of the tangent space t p f we also show that if every irreducible component of is of type a n or c n then the reduced tangent cones to x w 1 and x w 2 do not coincide as subvarieties of t p f

finding strong pseudoprimes to several bases ii define m to be the smallest strong pseudoprime to all the first m prime bases if we know the exact value of m we will have for integers n m a deterministic efficient primality testing algorithm which is easy to implement thanks to pomerance et al and jaeschke the m are known for 1 m 8 upper bounds for g 10 and 11 were first given by jaeschke and those for 10 and 11 were then sharpened by the first author in his previous paper math comp 70 2001 863 872 in this paper we first follow the first author s previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes spsp s n 1024 to the first five or six prime bases which have the form n pq with p q odd primes and q 1 k p 1 k 4 3 5 2 3 2 6 then we tabulate all carmichael numbers 1020 to the first six prime bases up to 13 which have the form n q1q2q3 with each prime factor qi 3 mod 4 there are in total 36 such carmichael numbers 12 numbers of which are also spsp s to base 17 5 numbers are spsp s to bases 17 and 19 one number is an spsp to the first 11 prime bases up to 31 as a result the upper bounds for 9 10 and 11 are lowered from 20 and 22 decimal digit numbers to a 19 decimal digit number 9 10 11 q11 3825 12305 65464 13051 19 digits 149491 747451 34233211 we conjecture that 9 10 11 3825 12305 65464 13051 and give reasons to support this conjecture the main idea for finding these carmichael numbers is that we loop on the largest prime factor q3 and propose necessary conditions on n to be a strong pseudoprime to the first 5 prime bases comparisons of effectiveness with arnault s bleichenbacher s jaeschke s and pinch s methods for finding carmichael numbers with three prime factors which are strong pseudoprimes to the first several prime bases are given

two kinds of strong pseudoprimes up to 10 36 let n 1 be an odd composite integer write n 1 2 s d with d odd if either b d 1 mod n or b 2r d 1 mod n for some r 0 1 s 1 then we say that n is a strong pseudoprime to base b or spsp b for short define t to be the smallest strong pseudoprime to all the first t prime bases if we know the exact value of t we will have for integers n 10 36 and to give reasons and numerical evidence of k2 and c 3 spsp s 12 where t resp t is the smallest k2 resp c 3 strong pseudoprime to all the first t prime bases for this purpose we describe procedures for computing and enumerating the two kinds of spsp s 10 36 to the first 9 prime bases the entire calculation took about 4000 hours on a pc pentium iv 1 8ghz recall that a k2 spsp is an spsp of the form n pq with p q primes and q 1 2 p 1 and that a c 3 spsp is an spsp and a carmichael number of the form n q 1 q 2 q 3 with each prime factor q i 3 mod 4

finding c 3 strong pseudoprimes let q 1 10 9 and so that 12 3186 65857 83403 11511 67461 24 digits 399165290221 798330580441 which was found by the author in an earlier paper we give reasons to support the conjecture the main idea of our method for finding those 21978 c 3 spsp 2 s is that we loop on candidates of signatures and kernels with heights bounded subject those candidates n q 1 q 2 q 3 of c 3 spsp 2 s and their prime factors q 1 q 2 q 3 to miller s tests and obtain the desired numbers at last we speed our algorithm for finding larger c 3 spsp s say up to 10 50 with a given signature to more prime bases comparisons of effectiveness with arnault s and our previous methods for finding c 3 strong pseudoprimes to the first several prime bases are given

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

primality testing and jacobi sums we present a theoretically and algorithmically simplified version of a primality testing algorithm that was recently invented by adleman and rumely the new algorithm performs well in practice it is the first pnmality test in existence that can routinely handle numbers of hundreds of decimal digits

strong pseudoprimes to the first eight prime bases define m to be the smallest strong pseudoprime to the firstm prime bases the exact value of m is known for 1 m 8 z zhang have found a 19decimal digit number q11 3825 12305 65464 13051 which is a strong pseudoprime to the first 11 prime bases and he conjectured that 9 10 11 q11 we prove the conjecture by algorithms

a lookup conjecture for rational smoothness the purpose of this paper is to present a new combinatorial criterion for rational smooth ness of a point in a schubert variety the main result proves the criterion for schubert varieties associated to partial flag varieties of the form g p where g is semisimple of classical type and p is a maximal parabolic subgroup in the case where g p is of cominuscule type not necessarily clas sical we prove a stronger result we conjecture that the criterion is valid for all schubert varieties the paper also contains some general results related to rational smoothness in schubert varieties 1 introduction the purpose of this paper is to present a new combinato rial criterion for rational smoothness of a point in a schubert variety the main result proves the criterion for schubert varieties associated to certain partial flag varieties we conjecture that it is true in general the paper also contains some general results related to rational smoothness in schubert varieties we now explain our criterion in more detail let g be a complex semisimple

average case error estimates for the strong probable prime test consider a procedure that chooses k bit odd numbers independently and from the uniform distribution subjects each number to t independent iterations of the strong probable prime test miller rabin test with randomly chosen bases and outputs the first number found that passes all t tests let pk t denote the probability that this procedure returns a composite number we obtain numerical upper bounds for pk t for various choices of k t and obtain clean explicit functions that bound pk t for certain infinite classes of k t for example we show pioo 10 2 in addition we characterize the worst case numbers with unusually many false witnesses and give an upper bound on their distribution that is probably close to best possible

limits of tame automorphisms of k x1 xn abstract let j consist of all endomorphisms of s k x1 xn whose jacobtan determinant det x i j is a non zero constant let g be the group of tame automorphisms of s in this paper we prove that g is dense in j in the formal power series topology in particular every automorphism of s is a limit of tame automorphisms we also demonstrate that certain subquotients of g have a natural lie algebra structure

on the combinatorics of b b orbits on group compactifications abstract it is shown that there is an order isomorphism from the poset v of b b orbits on the wonderful compactification of a semi simple adjoint group g with weyl group w to an interval in reverse chevalley bruhat order on a non canonically associated coxeter group w in general neither finite nor affine moreover preserves the corresponding kazhdan lusztig polynomials springer s partly conjectural construction of kazhdan lusztig polynomials for the analogues of v for general coxeter groups w is completed by reducing it by a similar order isomorphism to known results involving a twisted chevalley bruhat order on w

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

some new kinds of pseudoprimes we define some new kinds of pseudoprimes to several bases which generalize strong pseudoprimes we call them sylow p pseudoprimes and elementary abelian p pseudoprimes it turns out that every n 10 12 which is a strong pseudoprime to bases 2 3 and 5 is not a sylow p pseudoprime to two of these bases for an appropriate prime p n 1 we also give examples of strong pseudoprimes to many bases which are not sylow p pseudoprimes to two bases only where p 2 or 3

new stably tame automorphisms of polynomial algebras abstract let k x y k x1 xn y1 ym be the polynomial algebra in m n variables over a field k of characteristic 0 let be a locally nilpotent derivation of k x y such that yi 0 i 1 m and let act as a k y affine transformation over the free k y module freely generated by x1 xn we prove that the automorphism exp w of k x y is stably tame for every polynomial w from the kernel ker of this result is applied to the automorphisms of the polynomial algebra in five variables introduced recently by drensky and gupta and arising from wild automorphisms of generic matrix algebras we also give an algorithm for constructing new stably tame automorphisms in any number of variables and hence new potential candidates for wild automorphisms

verifying the goldbach conjecture up to 4 x 10 14 using a carefully optimized segmented sieve and an efficient checking algorithm the goldbach conjecture has been verified and is now known to be true up to 4 10 14 the program was distributed to various workstations it kept track of maximal values of the smaller prime p in the minimal partition of the even numbers where a minimal partition is a representation 2n p q with 2n p being composite for all p p the maximal prime p needed in the considered interval was found to be 5569 and is needed for the partition 389965026819938 5569 389965026814369

a one parameter quadratic base version of the baillie psw probable prime test the well known baillie psw probable prime test is a combination of a rabin miller test and a true i e with d n 1 lucas test arnault mentioned in a recent paper that no precise result is known about its probability of error grantham recently provided a probable prime test rqft with probability of error less than 1 7710 and pointed out that the lack of counter examples to the baillie psw test indicates that the true probability of error may be much lower in this paper we first define pseudoprimes and strong pseudoprimes to quadratic bases with one parameter tu t mod t2 ut 1 and define the base counting functions b n u 0 u n n is a psp tu and sb n u 0 u n n is an spsp tu then we give explicit formulas to compute b n and sb n and prove that for odd composites n b n n 2 and sb n n 8 and point out that these are best possible finally based on one parameter quadratic base pseudoprimes we provide a probable prime test called the one parameter quadratic base test opqbt which passed by all primes 5 and passed by an odd composite n p1r1 p2r2 psrs p1 p2 ps odd primes with probability of error n we give explicit formulas to compute n and prove that n 1 n4 3 for n nonsquare free with s 1 1 n2 3 for n square free with s 2 1 n2 7 for n square free with s 3 1 8s 4 166 p1 1 for n square free with s even 4 1 16s 5 119726 for n square free with s odd 5 1 4s i 1s 1 pi2 ri 1 otherwise i e for n nonsquare free with s 2 the running time of the opqbt is asymptotically 4 times that of a rabin miller test for worst cases but twice that of a rabin miller test for most composites we point out that the opqbt has clear finite group field structure and nice symmetry and is indeed a more general and strict version of the baillie psw test comparisons with gantham s rqft are given

the nil hecke ring and singularity of schubert varieties let g be a semi simple simply connected complex algebraic group and t b a maximal torus and a borel subgroup respectively let lie t be the cartan subalgebra of the lie algebra lie g and w n t t the weyl group associated to the pair g t where n t is the normalizer of t in g we can view any element omega overline omega mod t i w as the element denoted by the corresponding german character xo of g b denned as m overline omega b for any w i w there is associated the schubert variety rm x omega overline b omega b b subset g b and the t fixed points of x w under the canonical left action are precisely rm i omega left o upsilon in wand upsilon leqslant omega right

implementation of a new primality test an implementation of the cohen lenstra version see ibid vol 42 p 297 330 1984 of the adleman pomerance rumely primality test see l m adleman c pomerance and r s rumely ann of math vol 117 p 173 206 1983 is presented primality of prime numbers of up to 213 decimal digits can now routinely be proved within approximately ten minutes

on strong pseudoprimes to several bases with y k denoting the smallest strong pseudoprime to all of the first k primes taken as bases we determine the exact values for 5 q6 q7 q8 and give upper bounds for v 9 w t 1 we discuss the methods and underlying facts for obtaining these results 1 primality tests by means of strong pseudoprimes computer algebra systems as for instance axiom 2 use strong pseudoprimes for testing primality of integers the advantage of such tests is that they are very efficient the disadvantage is that they are only probabilistic tests when the integers are not restricted to certain intervals to make such tests deterministic for integers in prescribed intervals one has to know the exact number of necessary so called strong pseudoprimality tests for this purpose we introduce the numbers v1i v 2 for which we compute lower and upper bounds these numbers are defined and discussed in this section in 2 we derive some facts which are the basis for finding bounds for the numbers v k in 3 we discuss the methods which led to our results in view of fermat s little theorem we know that n is certainly not a prime when we have bn 1 i 1 mod n for an integer b with 1 0 and when n is a composite number then n is called a strong pseudoprime to base b if either

new automorphisms of generic matrix algebras and polynomial algebras abstract let t be the generic trace algebra generated by the algebra r of two generic 2 2 matrices and by all traces of the matrices from r over a field k we construct new automorphisms of t and r they induce automorphisms of the polynomial algebra in five variables which fix two of the variables our automorphisms of r are wild we do not know if the new automorphisms of k x 1 x 5 are tame when char k 2 we give the explicit form of all the constructed automorphisms

finding strong pseudoprimes to several bases define m to be the smallest strong pseudoprime to all the first m prime bases if we know the exact value of m we will have for integers n m a deterministic primality testing algorithm which is not only easier to implement but also faster than either the jacobi sum test or the elliptic curve test thanks to pornerance et al and jaeschke m are known for 1 m 8 upper bounds for 9 10 and 1 were given by jaeschke in this paper we tabulate all strong pseudoprimes spsp s n 10 24 to the first ten prime bases 2 3 29 which have the form n pq with p q odd primes and q 1 k p 1 k 2 3 4 there are in total 44 such numbers six of which are also spsp 31 and three numbers are spsp s to both bases 31 and 37 as a result the upper bounds for 10 and 11 are lowered from 28 and 29 decimal digit numbers to 22 decimal digit numbers and a 24 decimal digit upper bound for 12 is obtained the main tools used in our methods are the biquadratic residue characters and cubic residue characters we propose necessary conditions for n to be a strong pseudoprime to one or to several prime bases comparisons of effectiveness with both jaeschke s and arnault s methods are given