Prime Integers Research Articles

A Polynomial Analogue of the 3n + 1 Problem

viewed as mapping the integers to itself. The famous 3n + I conjecture postulates that after a finite number of iterations one always arrives at the value 1. This has been shown to be true for all n < 240. Whether it converges to 1 for every positive integer n is an open problem, see Lagarias [3]. The intricate relationship between the ring Z of integers and the polynomial ring Fq[x] of polynomials in a single variable x over the finite field Fq has been studied extensively. This discussion has included many topics such as comparisons between prime integers and irreducible polynomials, Goldbach type problems in both the in teger and polynomial settings (Effinger and Hayes [1]), and the study of twin primes and twin irreducibles and their densities, see [2]. In [2], the authors allude to the fact that in many (but not all) cases an analogous problem in the polynomial setting may be easier to resolve than the original problem. The result presented here indeed provides an example of that phenomenon. We start with the binary field F2 since this is the case most analogous to the integer version, see [2]. In the integer case, the first two primes are of course 2 and 3. In the F2[x] setting, the first two irreducible polynomials are x and x + 1. The analogue to the 3n + 1 problem for polynomials is the map Ci(f(x)) (x + l)f(x) + l if /(0)^0 fix) JK } if/(0) = 0 x

Discovering Binomial Identities with PascGaloisJE

We describe exercises in which students use PascGaloisJE to formulate conjectures about certain binomial identities which hold when the binomial coefficients are interpreted as elements in the cyclic group of integers modulo a prime integer p. In addition to having an appealing visual component, these exercises are open-ended and expose students to the investigative nature of mathematical discovery.

ON 2-ADJACENCY RELATION OF TWO-BRIDGE KNOTS AND LINKS

AbstractWe give a necessary condition for a two-bridge knot or link S(p,q) to be 2-adjacent to another two-bridge knot or link S(r,s). In particular, we show that if the trivial knot or link is 2-adjacent to S(p,q), then S(p,q) is trivial, that if S(p,q) is 2-adjacent to its mirror image, then S(p,q) is amphicheiral, and that for a prime integer p, if S(p,q) is 2-adjacent to S(r,s), then S(p,q)=S(r,s) or S(r,s)=S(1,0).

EXISTENCE AND STABILITY OF PERIODIC ORBITS OF PERIODIC DIFFERENCE EQUATIONS WITH DELAYS

In this paper, we investigate the existence and stability of periodic orbits of the p-periodic difference equation with delays xn = f(n - 1, xn-k). We show that the periodic orbits of this equation depend on the periodic orbits of p autonomous equations when p divides k. When p is not a divisor of k, the periodic orbits depend on the periodic orbits of gcd(p, k) nonautonomous p/gcd(p, k)-periodic difference equations. We give formulas for calculating the number of different periodic orbits under certain conditions. In addition, when p and k are relatively prime integers, we introduce what we call the pk-Sharkovsky's ordering of the positive integers, and extend Sharkovsky's theorem to periodic difference equations with delays. Finally, we characterize global stability and show that the period of a globally asymptotically stable orbit must be divisible by p.

Efficient Revocable Group Signature Schemes Using Primes

Group signature schemes with membership revocation have been intensively researched. In this paper, we propose revocable group signature schemes with less computational costs for signing/verification than an existing scheme without the update of the signer's secret keys. The key idea is the use of a small prime number embedded in the signer's secret key. By using simple prime integer relations between the secret prime and the public product of primes for valid or invalid members, the revocation is efficiently ensured. To show the practicality, we implemented the schemes and measured the signing/verification times in a common PC (Core2 DUO 2.13GHz). The times are all less than 0.5 seconds even for relatively large groups (10, 000 members), and thus our schemes are sufficiently practical.

Self-calibration of divided circles on the basis of a prime factor algorithm

A new method for the self-calibration of divided circles is presented which is based on a known prime factor algorithm for the discrete Fourier transform (DFT). The method, called prime factor division (PFD) calibration, is of interest in angle metrology specially for self-calibrating angle encoders, and generally for a significant shortening of the cross-calibration between two divided circles. It requires that the circular division number N can be expressed as a product N = R × S, whereby the factors R and S are relatively prime integer numbers. For the self-calibration of a divided circle, N difference measurements between R angle positions in a regular distribution and one reference angle position determined by S are evaluated by a two-dimensional DFT, yielding the N absolute division errors. The factor R is preferably chosen small, down to a minimum of R = 2, whereas the factor S may be as large as appropriate for the division number N of interest. In the case of a cross-calibration between two divided circles, the PFD method reduces the number of measurements necessary from N2 to (R + 1) × N. Experimental results are demonstrated for the calibrations of an optical polygon with 24 faces (prime factor product 3 × 8) and a gearwheel with 44 teeth (prime factor product 4 × 11).

Elasticity in certain block monoids via the Euclidean table

Abstract This paper continues the study begun in [GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤn, S) where S = $$\{ \bar 1,\bar a\} $$ (hereafter denoted ℬa(n)). We introduce in Section 2 the notion of a Euclidean table and show in Theorem 2.8 how it can be used to identify the irreducible elements of ℬa(n). In Section 3 we use the Euclidean table to compute the elasticity of ℬa(n) (Theorem 3.4). Section 4 considers the problem, for a fixed value of n, of computing the complete set of elasticities of the ℬa(n) monoids. When n = p is a prime integer, Proposition 4.12 computes the three smallest possible elasticities of the ℬa(p).

Lower bounds for numbers of ABC-hits

By an ABC-hit, we mean a triple ( a , b , c ) of relatively prime positive integers such that a + b = c and rad ( a b c ) < c . Denote by N ( X ) the number of ABC-hits ( a , b , c ) with c ⩽ X . In this paper we discuss lower bounds for N ( X ) . In particular we prove that for every ϵ > 0 and X large enough N ( X ) ⩾ exp ( ( log X ) 1 / 2 − ϵ ) .

Representation functions of bases for binary linear forms

Let $F(x_1,\ldots,x_m) = u_1 x_1 + \cdots + u_mx_m$ be a linear form with nonzero, relatively prime integer coefficients $u_1, \ldots, u_m$. For any set $A$ of integers, let $F(A)=\{F(a_1,\ldots,a_m): a_i \in A for i=1,\ldots,m\}.$ The {\it representation function} associated with the form $F$ is $$ R_{A,F}(n) = \card ( \{ (a_1,\ldots,a_m)\in A^m: F(a_1,\ldots, a_m) = n \} ). $$ The set $A$ is a {\it basis with respect to $F$ for almost all integers} if the set ${\bf Z} \setminus F(A)$ has asymptotic density zero. Equivalently, the representation function of a basis for almost all integers is a function $f:{\bf Z} \rightarrow {\bf N_0}\cup\{\infty\}$ such that $f^{-1}(0)$ has density zero. Given such a function, the inverse problem for bases is to construct a set $A$ whose representation function is $f$. In this paper the inverse problem is solved for binary linear forms. for binary linear forms.

On the hole index of L(2,1)-labelings of r-regular graphs

An L(2,1)-labeling of a graph G is an assignment of nonnegative integers to the vertices of G so that adjacent vertices get labels at least distance two apart and vertices at distance two get distinct labels. A hole is an unused integer within the range of integers used by the labeling. The lambda number of a graph G, denoted λ ( G ) , is the minimum span taken over all L(2,1)-labelings of G. The hole index of a graph G, denoted ρ ( G ) , is the minimum number of holes taken over all L(2,1)-labelings with span exactly λ ( G ) . Georges and Mauro [On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208–223] conjectured that if G is an r-regular graph and ρ ( G ) ⩾ 1 , then ρ ( G ) must divide r. We show that this conjecture does not hold by providing an infinite number of r-regular graphs G such that ρ ( G ) and r are relatively prime integers.

A search for primes from lesser primes

The purpose of this paper is to show that there is an algorithm which permits to define new primes from lesser already known primes. The method is based on the definition of “compact” sets of primes, and on the fundamental property, i.e., the fact that the sum or difference of two mutually prime integers have no common decomposition factors with these two mutually prime integers.

Frobenius Problem and dead ends in integers

Let a and b be positive and relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In addition, for given integers a and b, we explicitly describe all such elements in Z . Finally, we show that Z has only finitely many dead ends with respect to any finite symmetric generating set. In Appendix A we show that every finitely generated group has a generating set with respect to which dead ends exist.

An inequality concerning the elasticity of Krull monoids with divisor class group $\mathbb{Z}_p$

Let p be a prime integer and M a Krull monoid with divisor class group $\mathbb{Z}_p$ . We represent by S the set of nontrivial divisor classes of $\mathbb{Z}_p$ which contain prime divisors. We present a new inequality for the elasticity of M (denoted ρ (M)) which is dependent on the cardinality of S and argue that this inequality is the best possible. If M as above has | S| = 3, then it is known that $\rho(M)\in \{p/2, p/3,\ldots ,p/(\lfloor p/3\rfloor + 1)\}$ , but for large p, not all the values in this containment set can be realized. For each | S| = 3, we produce a submonoid $\widetilde{\mathcal{B}}(\mathbb{Z}_p,S)$ of $\mathcal{B}(\mathbb{Z}_p,S)$ such that $$ \{\,\rho(\widetilde{\mathcal{B}}(\mathbb{Z}_p,S))\,\mid \, S\subseteq \mathbb{Z}_p-\{0\}\mbox{ and }\mid S\mid =3\}=\{p/2, p/3,\ldots ,p/(\lfloor p/3\rfloor + 1)\}. $$ Keywords: Krull monoid, Block monoid, Elasticity of factorization Mathematics Subject Classification (2000): 20M14, 20D60, 13F05

On the distribution of points on multidimensional modular hyperbolas

We study the distribution of points on the $(n+1)$-dimensional modular hyperbola $a_1\cdots a_{n+1} \equiv c \pmod q$, where $q$ and $c$ are relatively prime integers. In particular, we show that an elementary argument leads to a straight-forward proof of a recent result of T.~Zhang and W.~Zhang, with a stronger error term. We also use character sums to obtain an asymptotic formula for the number of points in a given box that lie on such hyperbolas.

Frobenius Problem and the Covering Radius of a Lattice

Let $N \geq2$ and let $1 < a_1 < \cdots < a_N$ be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as $\sum_{i=1}^N a_i x_i {\rm where } x_1,\ldots,x_N$ are non-negative integers. The condition that $\gcd(a_1,\ldots,a_N)=1$ implies that such a number exists. The general problem of determining the Frobenius number given N and $a_1,\ldots,a_N$ is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.

Classification of homogeneous locally nilpotent derivations of k[X,Y,Z] Part I: Positive gradings of positive type

Let k be a field of characteristic zero, let a,b,c be relatively prime positive integers, and define a grading "g'' on the polynomial ring B = k[X,Y,Z] by declaring that X,Y,Z are homogeneous of degrees a,b,c, respectively. Consider the problem of classifying g-homogeneous locally nilpotent derivations of B. The present paper solves the case where g has positive type, which means that a,b,c are not pairwise relatively prime. The case where a,b,c are pairwise relatively prime is solved in our subsequent paper.

Multiscale Haar Orthonormal Matrices with the Corresponding Riesz Products and a Characterization of Cantor-Type Languages

We introduce a class of multiscale orthonormal matrices H(m) of order m×m, m = 2, 3,... . For m = 2 N, N = 1, 2,..., we get the well known Haar wavelet system. The term "multiscale" indicates that the construction of H(m) is achieved in different scales by an iteration process, determined through the prime integer factorization of m and by repetitive dilation and translation operations on matrices. The new Haar transforms allow us to detect the underlying ergodic structures on a class of Cantor-type sets or languages. We give a sufficient condition on finite data of lengthm, or step functions determined on the intervals [k/m, (k + 1)/m) , k = 0,...,m − 1 of [0, 1), to be written as a Riesz-type product in terms of the rows of H(m). This allows us to approximate in the weak-* topology continuous measures by Riesz-type products.

Congruences for sums of binomial coefficients

Let q > 1 and m > 0 be relatively prime integers. We find an explicit period ν m ( q ) such that for any integers n > 0 and r we have [ n + ν m ( q ) r ] m ( a ) ≡ [ n r ] m ( a ) ( mod q ) whenever a is an integer with gcd ( 1 − ( − a ) m , q ) = 1 , or a ≡ − 1 ( mod q ) , or a ≡ 1 ( mod q ) and 2 | m , where [ n r ] m ( a ) = ∑ k ≡ r ( mod m ) ( n k ) a k . This is a further extension of a congruence of Glaisher.

Two-parameter families of strange attractors

Periodically driven two-dimensional nonlinear oscillators can generate strange attractors that are periodic. These attractors are mapped in a locally 1-1 way to entire families of strange attractors that are indexed by a pair of relatively prime integers (n(1),n(2)), with n(1)>/=1. The integers are introduced by imposing periodic boundary conditions on the entire strange attractor rather than individual trajectories in the attractor. The torsion and energy integrals for members of this two parameter family of locally identical strange attractors depend smoothly on these integers.

On the computation of representations of primes as sums of four squares

Lagrange proved that every positive integer is the sum of four squares of natural numbers. Although Lagrange’s proof is constructive, it is not known whether the relative algorithm produces a four-square representation of any prime or non-prime integer with deterministic polynomial complexity. Limited to prime numbers, it is proved that their representation as the sums of four squares can be obtained with deterministic polynomial complexity. The key to this computational accomplishment is the evaluation of square roots modulo a prime, through Schoof’s method for counting the number of points on elliptic curves over prime fields.