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) to the first five or six prime bases, which have the form with odd primes and ; then we tabulate all Carmichael numbers , to the first six prime bases up to 13, which have the form with each prime factor . 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 and are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number:
We conjecture that
and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor and propose necessary conditions on to be a strong pseudoprime to the first 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.
The main purpose of this paper is to give exact values of for ; to give a lower bound of : ; and to give reasons and numerical evidence of K2- and -spsp's to support the following conjecture: for any , where (resp. ) is the smallest K2- (resp. -) strong pseudoprime to all the first prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp's 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: with primes and ; and that a -spsp is an spsp and a Carmichael number of the form: with each prime factor mod .)
This paper concerns the Rayleigh-Ritz method for computing an approximation to an eigenspace of a general matrix from a subspace that contains an approximation to . The method produces a pair that purports to approximate a pair , where is a basis for and . In this paper we consider the convergence of as the sine of the angle between and approaches zero. It is shown that under a natural hypothesis--called the uniform separation condition--the Ritz pairs converge to the eigenpair . When one is concerned with eigenvalues and eigenvectors, one can compute certain refined Ritz vectors whose convergence is guaranteed, even when the uniform separation condition is not satisfied. An attractive feature of the analysis is that it does not assume that has distinct eigenvalues or is diagonalizable.
Let be a product of two distinct primes and . We show that for almost all exponents with the RSA pairs are uniformly distributed modulo when runs through
- the group of units modulo (that is, as in the classical RSA scheme);
- the set of -products , , where are selected at random (that is, as in the recently introduced RSA scheme with precomputation).
In this paper, we enumerate all number fields of degree of discriminant smaller than in absolute value containing a quintic field having one real place. For each one of the (resp. found fields of signature (resp. the field discriminant, the quintic field discriminant, a polynomial defining the relative quadratic extension, the corresponding relative discriminant, the corresponding polynomial over , and the Galois group of the Galois closure are given.
In a supplementary section, we give the first coincidence of discriminant of (resp. nonisomorphic fields of signature (resp. .
Let be an even integer, . The resultant of the polynomials and is known as Wendt's determinant of order . We prove that among the prime divisors of only those which divide or can be larger than , where and is the th Lucas number, except when and . Using this estimate we derive criteria for the nonsolvability of Fermat's congruence.
Consider the pseudorandom number generator where we are given the modulus , the initial value and the exponent . One case of particular interest is when the modulus is of the form , where are different primes of the same magnitude. It is known from work of the first and third authors that for moduli , if the period of the sequence exceeds , then the sequence is uniformly distributed. We show rigorously that for almost all choices of it is the case that for almost all choices of , the period of the power generator exceeds . And so, in this case, the power generator is uniformly distributed.
We also give some other cryptographic applications, namely, to ruling-out the cycling attack on the RSA cryptosystem and to so-called time-release crypto.
The principal tool is an estimate related to the Carmichael function , the size of the largest cyclic subgroup of the multiplicative group of residues modulo . In particular, we show that for any , we have for all integers with , apart from at most exceptions.
Some years ago, compactly supported divergence-free wavelets were constructed which also gave rise to a stable (biorthogonal) wavelet splitting of . These bases have successfully been used both in the analysis and numerical treatment of the Stokes and Navier-Stokes equations. In this paper, we construct stable wavelet bases for the stream function spaces . Moreover, -free vector wavelets are constructed and analysed. The relationship between and are expressed in terms of these wavelets. We obtain discrete (orthogonal) Hodge decompositions.
Our construction works independently of the space dimension, but in terms of general assumptions on the underlying wavelet systems in that are used as building blocks. We give concrete examples of such bases for tensor product and certain more general domains . As an application, we obtain wavelet multilevel preconditioners in and .
We examine the problem of factoring the th cyclotomic polynomial, over , and distinct primes. Given the traces of the roots of we construct the coefficients of in time . We demonstrate a deterministic algorithm for factoring in time when has precisely two irreducible factors. Finally, we present a deterministic algorithm for computing the sum of the irreducible factors of in time .
For any , let be the th prime number. In this paper, we confirm a conjecture of Erdos and Stewart concerning all the solutions of the diophantine equation , when .
Consider a differential equation with and , where is a Lie algebra of the matricial Lie group . Every can be mapped to by the matrix exponential map with .
Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation of the exact solution , , by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value . This ensures that .
When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby is approximated by a product of simpler exponentials.
In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of and are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper.
We consider sequences of matrices with a block structure spectrally distributed as an -variate matrix-valued function , and, for any , we suppose that is a linear and positive operator. For every fixed we approximate the matrix in a suitable linear space of matrices by minimizing the Frobenius norm of when ranges over . The minimizer is denoted by . We show that only a simple Korovkin test over a finite number of polynomial test functions has to be performed in order to prove the following general facts:
- 1.
- the sequence is distributed as ,
- 2.
- the sequence is distributed as the constant function (i.e. is spectrally clustered at zero).
The present paper is a continuation of an earlier work by the author. We propose some new definitions of -adic continued fractions. At the end of the paper we give numerical examples illustrating these definitions. It turns out that for every if then has a periodic continued fraction expansion. The same is not true in for some larger values of
with probability . Finding the number involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal measure, a computer calculation, and a rounding error analysis to validate the computer calculation.