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.
For each prime , let be the product of the primes less than or equal to . We have greatly extended the range for which the primality of and are known and have found two new primes of the first form ( ) and one of the second (). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found.
Suppose is a finite-dimensional linear space based on a triangulation of a domain , and let denote the -projection onto . Provided the mass matrix of each element and the surrounding mesh-sizes obey the inequalities due to Bramble, Pasciak, and Steinbach or that neighboring element-sizes obey the global growth-condition due to Crouzeix and Thomée, is -stable: For all we have with a constant that is independent of, e.g., the dimension of .
This paper provides a more flexible version of the Bramble-Pasciak- Steinbach criterion for -stability on an abstract level. In its general version, (i) the criterion is applicable to all kind of finite element spaces and yields, in particular, -stability for nonconforming schemes on arbitrary (shape-regular) meshes; (ii) it is weaker than (i.e., implied by) either the Bramble-Pasciak-Steinbach or the Crouzeix-Thomée criterion for regular triangulations into triangles; (iii) it guarantees -stability of a priori for a class of adaptively-refined triangulations into right isosceles triangles.
As in the earlier paper with this title, we consider a question of Byrnes concerning the minimal length of a polynomial with all coefficients in which has a zero of a given order at . In that paper we showed that for all and showed that the extremal polynomials for were those conjectured by Byrnes, but for that rather than . A polynomial with was exhibited for , but it was not shown there that this extremal was unique. Here we show that the extremal is unique. In the previous paper, we showed that is one of the 7 values or . Here we prove that without determining all extremal polynomials. We also make some progress toward determining . As in the previous paper, we use a combination of number theoretic ideas and combinatorial computation. The main point is that if is a primitive th root of unity where is a prime, then the condition that all coefficients of be in , together with the requirement that be divisible by puts severe restrictions on the possible values for the cyclotomic integer .
We prove the stability in of the projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions. We give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions. In particular, stability of the projection in holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically.
Let be the characteristic polynomial of the Hecke operator acting on the space of level 1 cusp forms . We show that is irreducible and has full Galois group over for and , prime.
Directional Newton methods for functions of variables are shown to converge, under standard assumptions, to a solution of . The rate of convergence is quadratic, for near-gradient directions, and directions along components of the gradient of with maximal modulus. These methods are applied to solving systems of equations without inversion of the Jacobian matrix.
A set of primes involving numbers such as , where and , is defined. An algorithm for computing discrete logs in the finite field of order with is suggested. Its heuristic expected running time is for , where as , , and . At present, the most efficient algorithm for computing discrete logs in the finite field of order for general is Schirokauer's adaptation of the Number Field Sieve. Its heuristic expected running time is for . Using rather than general does not enhance the performance of Schirokauer's algorithm. The definition of the set and the algorithm suggested in this paper are based on a more general congruence than that of the Number Field Sieve. The congruence is related to the resultant of integer polynomials. We also give a number of useful identities for resultants that allow us to specify this congruence for some .
Although the quantity is known for some Pisot numbers , there has been no general method for computing . This paper gives such an algorithm. With this algorithm, some properties of and its generalizations are investigated.
A related question concerns the analogy of , denoted , where the coefficients are restricted to ; in particular, for which non-Pisot numbers is nonzero? This paper finds an infinite class of Salem numbers where .
A systematic search for optimal lattice rules of specified trigonometric degree over the hypercube has been undertaken. The search is restricted to a population of lattice rules . This includes those where the dual lattice may be generated by points for each of which . The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for , and , , a list of -optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed.
By a prime gap of size , we mean that there are primes and such that the numbers between and are all composite. It is widely believed that infinitely many prime gaps of size exist for all even integers . However, it had not previously been known whether a prime gap of size existed. The objective of this article was to be the first to find a prime gap of size , by using a systematic method that would also apply to finding prime gaps of any size. By this method, we find prime gaps for all even integers from to , and some beyond. What we find are not necessarily the first occurrences of these gaps, but, being examples, they give an upper bound on the first such occurrences. The prime gaps of size listed in this article were first announced on the Number Theory Listing to the World Wide Web on Tuesday, April 8, 1997. Since then, others, including Sol Weintraub and A.O.L. Atkin, have found prime gaps of size with smaller integers, using more ad hoc methods. At the end of the article, related computations to find prime triples of the form , , and their application to divisibility of binomial coefficients by a square will also be discussed.
Let 2$">, an -th primitive root of 1, mod a prime number, a primitive root modulo and . We study the Jacobi sums , , where is the least nonnegative integer such that mod . We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families , , of irreducible polynomials of Gaussian periods, , of degree , where is a suitable set of primes mod . We exhibit examples of such families for several small values of , and give a MAPLE program to construct more of them.