First-hit analysis of algorithms for computing quadratic irregularity

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are uniformly distributed modulo for every . This is the basis of a well-known heuristic, given by Siegel, estimating the frequency of irregular primes. So far, analyses have shown that if is a real quadratic field, then the values of the zeta function at negative odd integers are also distributed as expected modulo for any . We use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude. We also discuss alternative ways of collecting large amounts of data to test the heuristic.

     

Comparison of algorithms to calculate quadratic irregularity of prime numbers

In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields , using the values of the zeta function at negative integers as our ``higher Bernoulli numbers'. In the case where is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (The author would like to thank Henri Cohen for suggesting an analysis of the second formula.) We briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of -irregularity (more generally, ``quadratic irregularity') of a prime number.

     

On the primality of

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.

     

How to Choose Secret Parameters for RSA-Type Cryptosystems over Elliptic Curves

Recently, and contrary to the common belief, Rivest and Silverman argued that the use of strong primes is unnecessary in the RSA cryptosystem. This paper analyzes how valid this assertion is for RSA-type cryptosystems over elliptic curves. The analysis is more difficult because the underlying groups are not always cyclic. Previous papers suggested the use of strong primes in order to prevent factoring attacks and cycling attacks. In this paper, we only focus on cycling attacks because for both RSA and its elliptic curve-based analogues, the length of the RSA-modulus n is typically the same. Therefore, a factoring attack will succeed with equal probability against all RSA-type cryptosystems. We also prove that cycling attacks reduce to find fixed points, and derive a factorization algorithm which (most probably) completely breaks RSA-type systems over elliptic curves if a fixed point is found.     

First Order Differential Operators Associated to the Cauchy—Riemann Operator in the Plane

The article deals with initial value problems of type δw/δt = Fw, w(0, ·) = φ where t is the time and F is a linear first order operator acting in the z = x ? iy-plane. In view of the classical Cauchy-Kovalevkaya Theorem, the initial value problem is solvable provided F has holomorphic coefficients and the initial function is holomorphic. On the other hand, the Lewy example [H. Lewy (1957). An example of a smooth linear partial differential equation without solution. Ann. of Math., 66, 155–158.] shows that there are equations of the above form with infinitely differentiable coefficients not having any solutions. The article in hand constructs, conversely, all linear operators F for which the initial value problem with an arbitrary holomorphic initial function is always solvable. In particular, we shall see that there are equations of that type whose coefficients are only continuous.     

Constitutive modeling of orthotropic sheet metals by presenting hardening-induced anisotropy

An essential work on the constitutive modeling of rolled sheet metals is the consideration of hardening-induced anisotropy. In engineering applications, we often use testing results of four specified experiments, three uniaxial-tensions in rolling, transverse and diagonal directions and one equibiaxial-tension, to describe the anisotropic features of rolled sheet metals. In order to completely take all these experimental results, including stress-components and strain-ratios, into account in the constitutive modeling for presenting hardening-induced anisotropy, an appropriate yield model is developed. This yield model can be characterized experimentally from the offset of material yield to the end of material hardening. Since this adaptive yield model can directly represent any subsequent yielding state of rolled sheet metals without the need of an artificially defined “effective stress”, it makes the constitutive modeling simpler, clearer and more physics-based. This proposed yield model is convex from the initial yield state till the end of strain-hardening and is well-suited in implementation of finite element programs.     

Replication in critical graphs and the persistence of monomial ideals

Motivated by questions about square-free monomial ideals in polynomial rings, in 2010 Francisco et al. conjectured that for every positive integer k and every k-critical (i.e., critically k  -chromatic) graph, there is a set of vertices whose replication produces a (k+1)$(k+1)$-critical graph. (The replication of a set W of vertices of a graph is the operation that adds a copy of each vertex w in W, one at a time, and connects it to w and all its neighbours.)     

On the Galois groups of Legendre polynomials

Ever since Legendre introduced the polynomials that bear his name in 1785, they have played an important role in analysis, physics and number theory, yet their algebraic properties are not well-understood. Stieltjes conjectured in 1890 how they factor over the rational numbers. In this paper, assuming Stieltjes’ conjecture, we formulate a conjecture about the Galois groups of Legendre polynomials, to the effect that they are “as large as possible,” and give theoretical and computational evidence for it.     

Selberg’s integral for Ramanujan automorphic representation

Let π _Δ be the automorphic representation of GL(2,ℚ_A) associated with Ramanujan modular form Δ and L(s, π _Δ) the global L-function attached to π _Δ. We study Selberg’s integral for the automorphic L-function L(s, π _Δ) under GRH. Our results give the information for the number of primes in short intervals attached to Ramanujan automorphic representation.     

关于Romanoff常数及其推广问题

1934年,Romanoff证明了：可表为一个素数和一个2的方幂之和的大奇数在全体正整数中具有正密度．本文证明了此密度大于0．09322,从而改进了该问题的已有结果0．0868．作为此问题的推广,本文还建立了一个类似的数值结果：可表为两个素数的平方和两个2的方幂之和的大偶数具有正密度．