On congruences of Euler numbers modulo powers of two

In this paper, we establish some identities involving the Euler numbers, the Euler numbers of order 2 and the central factorial numbers, and give a new proof of a classical result due to M.A. Stern.

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On recurrences for sums of powers of binomial coefficients

The recurrence for sums of powers of binomial coefficients is considered and a lower bound for the minimal length of the recurrence is obtained by using the properties of congruence.

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A thermodynamic classification of real numbers

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A new classification scheme for real numbers is given, motivated by ideas from statistical mechanics in general and work of Knauf (1993) [16] and Fiala and Kleban (2005) [8] in particular. Critical for this classification of a real number will be the Diophantine properties of its continued fraction expansion.

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Coleman maps for modular forms at supersingular primes over Lubin-Tate extensions

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Given an elliptic curve with supersingular reduction at an odd prime p, Iovita and Pollack have generalised results of Kobayashi to define even and odd Coleman maps at p over Lubin-Tate extensions given by a formal group of height 1. We generalise this construction to modular forms of higher weights.

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On the mean value of the index of composition of an integral ideal

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In this paper, we study the mean value of the index of composition of an integral ideal, and get an asymptotic formula for it.

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Distinct length modular zero-sum subsequences: A proof of Graham's conjecture

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Let S be a sequence of n nonnegative integers not exceeding n−1 such that S takes at least three distinct values. We show that S has two nonempty zero-sum subsequences with distinct lengths. This proves a conjecture of R.L. Graham. The validity of this conjecture was verified by Erd?s and Szemerédi for all sufficiently large prime n.

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Sun's conjectures on fourth powers in the class group of binary quadratic forms

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We prove five of Sun's conjectures on the index of the subgroup of fourth powers in the class group of binary quadratic forms.

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Expander graphs based on GRH with an application to elliptic curve cryptography

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We present a construction of expander graphs obtained from Cayley graphs of narrow ray class groups, whose eigenvalue bounds follow from the Generalized Riemann Hypothesis. Our result implies that the Cayley graph of ^∗(Z/qZ) with respect to small prime generators is an expander. As another application, we show that the graph of small prime degree isogenies between ordinary elliptic curves achieves nonnegligible eigenvalue separation, and explain the relationship between the expansion properties of these graphs and the security of the elliptic curve discrete logarithm problem.

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Estimating the Somos' quadratic recurrence constant

TextThe aim of this paper is to provide some estimates about the Somos' quadratic recurrence constant, using its relation with the generalized Euler constant.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3QjmHit3mC4.     

A covering system whose smallest modulus is 40

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Paul Erd?s, in 1950, asked whether for each positive integer N there exists a finite set of congruence classes, with distinct moduli, covering the integers, whose smallest modulus is N. In this vein, we construct a covering system of the integers with smallest modulus N=40.

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Endomorphism algebras of Jacobians of certain superelliptic curves

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Let p be a prime, and q a power of p. Using Galois theory, we show that over a field K of characteristic zero, the endomorphism algebras of the Jacobians of certain superelliptic curves yq=f(x) are products of cyclotomic fields.

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Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

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In a previous paper Najman (in press) [9], the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and . Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over Q(i) and , we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves over Q(i) and .

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Elliptic curve cryptography: The serpentine course of a paradigm shift

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Over a period of sixteen years elliptic curve cryptography went from being an approach that many people mistrusted or misunderstood to being a public key technology that enjoys almost unquestioned acceptance. We describe the sometimes surprising twists and turns in this paradigm shift, and compare this story with the commonly accepted Ideal Model of how research and development function in cryptography. We also discuss to what extent the ideas in the literature on “social construction of technology” can contribute to a better understanding of this history.

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On Jordan ideals and matrix rings

For A, a commutative ring, and results by Costa and Keller characterize certain -normalized subgroups of the symplectic group, via structures utilizing Jordan ideals and the notion of radices. The following work creates a Jordan ideal structure theorem for -graded rings, A₀A₁, and a -graded matrix algebra. The major theorem is a generalization of Costa and Keller’s previous work on matrix algebras over commutative rings.     

Effective equidistribution and the Sato-Tate law for families of elliptic curves

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Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate law. We present two methods of proof. Both use the framework of Murty and Sinha (2009) [MS]; the first involves only knowledge of the moments of the Fourier coefficients of the L-functions and combinatorics, and saves a logarithm, while the second requires a Sato-Tate law. Our purpose is to illustrate how the caliber of the result depends on the error terms of the inputs and what combinatorics must be done.

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Picard sequences with every orbit containing infinitely many perfect n-powers

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Finding a function which generates a sequence via iteration whose values at one or many points in its domain satisfy certain prescribed properties, i.e., finding a function such that the Picard orbit(s) of one or many points in its domain which possess some given properties, is an interesting problem. Given any positive integer n greater than one, we construct in this paper families of functions on the natural numbers such that the sequence of the iterations of each of these functions at any positive integer s contains infinitely many perfect n-powers. In terms of Picard sequences, this amounts to constructing a function whose Picard orbit at every point in its domain contains infinitely many perfect n-powers.

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Properties of the Beurling generalized primes

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In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that , where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit exists. We also show that this limit coincides with ; for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small |N(x)−[x]| can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that and conjecture that this is the best possible bound.

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Polynomial-time approximation schemes for piercing and covering with applications in wireless networks

Let be a set of disks of arbitrary radii in the plane, and let be a set of points. We study the following three problems: (i) Assuming contains the set of center points of disks in , find a minimum-cardinality subset of (if exists), such that each disk in is pierced by at least h points of , where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming is such that for each there exists a point in whose distance from D's center is at most αr(D), where r(D) is D's radius and 0α<1 is a given constant, find a minimum-cardinality subset of , such that each disk in is pierced by at least one point of . We call this problem minimum discrete piercing with cores. (iii) Assuming is the set of center points of disks in , and that each covers at most l points of , where l is a constant, find a minimum-cardinality subset of , such that each point of is covered by at least one disk of . We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178–189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a (1+ε)-approximation for minimum discrete piercing (i.e., for arbitrary ).     

Adaptive thinning algorithm of nonsingular triangulation

This paper studies adaptive thinning strategies for the non-singular triangulation of scattered data by C¹-rational spline function. Given a set of points in R², Luo, Liu and Chen have presented a triangulation algorithm which ensures the non-singularity of and spaces. In this paper, we improve the algorithm to reduce the number of knots of the triangulation within a given tolerance, while the non-singularity of and spaces is ensured. Our strategies presented here depend on both the locations of the data points in the plane, and the data values at these points. We give the definition of discrete norm for C¹-rational spline function by using its coefficients. Then a weight is assigned to each knot, which is a measure of the importance of knot in the representation of spline. When the weight of the knot is less than the given tolerance, its influence is regarded negligible, then it can be removed. It’s a discrete method. In the end of this paper several numerical examples are presented to show the feasibility and validity of our algorithm.     

The Bowman–Bradley theorem for multiple zeta-star values

TextThe Bowman–Bradley theorem asserts that the multiple zeta values at the sequences obtained by inserting a fixed number of twos between $3,1,\dots ,3,1$ add up to a rational multiple of a power of π. We establish its counterpart for multiple zeta-star values by showing an identity in a non-commutative polynomial algebra introduced by Hoffman.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=LpqA2OJ6vP8.