Browkin’s discriminator conjecture

Let q = p s be a power of a prime number p and let $${\mathbb {F}_{\rm q}}$$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over $${\mathbb {F}_{\rm q}}$$ . In particular we obtain the explicit factorization of the cyclotomic polynomial $${\Phi_{2^nr}}$$ over $${\mathbb {F}_{\rm q}}$$ where both r ? 3 and q are odd, gcd(q, r) = 1, and $${n\in \mathbb{N}}$$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of $${\Phi_r}$$ over $${\mathbb {F}_{\rm q}}$$ is given to us as a known. Let $${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$$ be the factorization of $${n \in \mathbb{N}}$$ into powers of distinct primes p i , 1 ? i ? s. In the case that the multiplicative orders of q modulo all these prime powers $${p_i^{e_i}}$$ are pairwise coprime, we show how to obtain the explicit factors of $${\Phi_{n}}$$ from the factors of each $${\Phi_{p_i^{e_i}}}$$ . We also demonstrate how to obtain the factorization of $${\Phi_{mn}}$$ from the factorization of $${\Phi_n}$$ when q is a primitive root modulo m and $${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$$ Here $${\phi}$$ is the Euler's totient function, and ord n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over $${\mathbb {F}_{\rm q}}$$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334---336, 1984).

Let p be an odd prime number and $$\ell $$ an odd prime number dividing $$p-1$$ . We denote by $$F=F_{p,\ell }$$ the real abelian field of conductor p and degree $$\ell $$ , and by $$h_F$$ the class number of F. For a prime number $$r \ne p,\,\ell $$ , let $$F_{\infty }$$ be the cyclotomic $$\mathbb {Z}_r$$ -extension over F, and $$M_{\infty }/F_{\infty }$$ the maximal pro-r abelian extension unramified outside r. We prove that $$M_{\infty }$$ coincides with $$F_{\infty }$$ and consequently $$h_F$$ is not divisible by r when r is a primitive root modulo $$\ell $$ and r is smaller than an explicit constant depending on p.

Linear complexity is an important cryptographic index of sequences. We study the linear complexity of $$p(q-1)$$ p ( q - 1 ) -periodic Legendre---Sidelnikov sequences, which combine the concepts of Legendre sequences and Sidelnikov sequences. We get lower and upper bounds on the linear complexity in different cases, and experiments show that the upper bounds can be attained. Remarkably, we associate the linear complexity of Legendre---Sidelnikov sequences with some famous primes including safe prime and Fermat prime. If $$2$$ 2 is a primitive root modulo $$\frac{q-1}{2}$$ q - 1 2 , and $$q$$ q is a safe prime greater than 7, the linear complexity is the period if $$p\equiv 3 \pmod 8$$ p ? 3 ( mod 8 ) ; $$p(q-1)-p+1$$ p ( q - 1 ) - p + 1 if $$p\equiv q \equiv 7 \pmod 8$$ p ? q ? 7 ( mod 8 ) , and $$p(q-1)-\frac{p-1}{2}$$ p ( q - 1 ) - p - 1 2 if $$p \equiv 7 \pmod 8, q \equiv 3 \pmod 8$$ p ? 7 ( mod 8 ) , q ? 3 ( mod 8 ) . If $$q$$ q is a Fermat prime, the linear complexity is the period if $$p \equiv 3 \pmod 8$$ p ? 3 ( mod 8 ) , and $$p(q-1)-q+2$$ p ( q - 1 ) - q + 2 if $$p \equiv 5 \pmod 8$$ p ? 5 ( mod 8 ) . It is very interesting that the Legendre---Sidelnikov sequence has maximal linear complexity and is balanced if we choose $$p=q$$ p = q to be some safe prime.

Arithmetic properties of some families in [Formula: see text] and hence in [Formula: see text] are obtained by using the cyclotomic classes of order 2 with respect to [Formula: see text], where [Formula: see text] is primitive root modulo [Formula: see text], [Formula: see text] and [Formula: see text]. The form of these cyclotomic classes enables us to generalize the results obtained in [S. Jain and S. Batra, Cyclotomy and arithmetic properties of some families in [Formula: see text], Asian-Eur. J. Math. 13(4) (2020) 2050077].

The explicit expression for the $$8n+6$$ primitive idempotents in $$FG$$ (the group algebra of the cyclic group $$G$$ of order $$8p^{n}$$ , where $$p$$ is an odd prime, $$n\ge 1)$$ over the finite field $$F$$ of prime power order $$q$$ , where $$q$$ is of the form $$8k+5$$ and is a primitive root modulo $$p^{n}$$ are obtained. The minimum distances, dimensions and the generating polynomials of the minimal cyclic codes generated by these primitive idempotents are also obtained.

If a knot K has Alexander polynomial different from 1, then its knot group, G maps onto some metacyclic group, Zr§Zp. We show that in that case, it also has a homomorphism onto a split extension of a free abelian group of rank p — 1 by Zr§Zp, and hence also onto a split extension of a direct sum of p — 1 cyclic groups of order s by the metacyclic group. In many cases, (such as if s is coprime with p), this group can be specified exactly. Otherwise there are a finite number of possibilities. A special case is Perko's result that a homomorphism of a knot group onto S3 = Z2^Z?) lifts to S4 = Z 2ίS)Z3<$(Z2 Θ Z2). As an application we obtain information about the derived series of G. In a final section it is shown how to associate a rational polynomial invariant to every metacyclic representation. 1. Lifting metacyclic representations. Let p be a prime, r a divisor of p — 1 and β a primitive rth root modulo p. Let E — Zj§Zp — (Y, S: γr zn sp = 19Y'1SY= Sβ). Up to isomorphism, the group is independent of β. Let G be the knot group of a knot, AT, in the 3-sphere, S3, and let φ be a homomoφhism of G onto E which takes a meridian, m, of K to YaSh. Then g.c.d. (α, r) — 1, since G is generated by conjugates of m. Setting X — YaSb and eliminating 7, we obtain a presentation

On [formula omitted]-packing colourings of distance graphs [formula omitted] and [formula omitted

For a nondecreasing sequence of integers $$S=(s_1, s_2, \ldots )$$ an S-packing k-coloring of a graph G is a mapping from V(G) to $$\{1, 2,\ldots ,k\}$$ such that vertices with color i have pairwise distance greater than $$s_i$$ . By setting $$s_i = d + \lfloor \frac{i-1}{n} \rfloor $$ we obtain a (d, n)-packing coloring of a graph G. The smallest integer k for which there exists a (d, n)-packing coloring of G is called the (d, n)-packing chromatic number of G. In the special case when d and n are both equal to one we obtain the packing chromatic number of G. We determine the packing chromatic number of base-3 Sierpinski graphs and provide new results on (d, n)-packing chromatic colorings for this class of graphs. By using a dynamic algorithm, we establish the packing chromatic number for H-graphs.

&lt;abstract&gt;&lt;p&gt;In the authors' earlier work, the SEL Egyptian fraction expansion for any real number was constructed and characterizations of rational numbers by using such expansion were established. These results yield a generalized version of the results for the Fibonacci-Sylvester and the Engel series expansions. Under a certain condition, one of such characterizations also states that the SEL Egyptian fraction expansion is finite if and only if it represents a rational number. In this paper, we obtain an upper bound for the length of the SEL Egyptian fraction expansion for rational numbers, and the exact length of this expansion for a certain class of rational numbers is verified. Using such expansion, not only is a large class of transcendental numbers constructed, but also an explicit bijection between the set of positive real numbers and the set of sequences of nonnegative integers is established.&lt;/p&gt;&lt;/abstract&gt;

In this article, we prove that the sequence consisting of quadratic non-residues which are not primitive root modulo a prime p obeys Poisson law whenever \(\displaystyle \frac{p-1}{2}- \phi (p-1)\) is reasonably large as a function of p. To prove this, we count the number of \(\ell \)-tuples of quadratic non-residues which are not primitive roots mod p, thereby generalizing one of the results obtained in Gun et al. (Acta Arith, 129(4):325–333, 2007, [8]).

On class numbers of algebraic function fields defined by y2 = xn + a over GF( p)

A gravitációs hullámok első közvetlen detektálása után a kihívás az, hogy a detektorok segítségével minél pontosabb képet kapjunk az ezen hullámokon keresztül megfigyelhető jelenségekről, vagyis megszülessen a gravitációshullám-csillagászat. Ebben kiemelt szerepe van a kompakt objektumok által alkotott kettős rendszereknek, mivel az általuk kibocsátott hullámok széles paramétertartományban nagy pontossággal leírhatók. Ehhez azonban szükségünk van a kettős rendszer mozgásának leírására. Ezt a poszt-newtoni sorfejtés alkalmazásával tesszük meg, figyelembe véve a pálya excentricitásának és a testek forgásának hatásait. A korábbi munkák tisztázzák a szeparációs vektor nagysága változásának részleteit, azonban a kibocsátott gravitációs hullámok leírásához szükségünk van a vektor-irányokat jellemző szögmennyiségek dinamikájának leírására is. Az ehhez szükséges egyenletek kerülnek jelen cikkben származtatásra, illetve részletezésre kerül ezek megoldásának módszerei. Célkitűzés továbbá a korábbi, alapanyagként használt publikációk hibáinak feltárása, javítása.

Clinical Summaries

A generalisation of the classical Halton sequence $$(\phi _{\beta }(n))_{n\in \mathbb {N}}$$ has emerged in recent years based on $$\beta $$ -adic expansions of elements of [0, 1). In the case where $$\beta $$ is a natural number greater than 1, this reduces to the classical Halton sequence. In this paper, we use ergodic and analytic methods to prove the uniform distribution of a sequence $$(\phi _{\beta }(k_j))_{j\in \mathbb {N}}$$ for the sequence of integers $$(k_j)_{j\ge 0}$$ , which is both Hartman uniformly distributed and good universal. This builds on earlier work of M. Hofer, M. R. Iaco and R. Tichy in the special case $$k_j =j \ (j=0,1, \ldots )$$ . Variants of this phenomenon are also studied.

Human epidermal growth factor receptors (HER)-also known as EGFR or ErbB receptors-are a subfamily of receptor tyrosine kinases (RTKs) that play crucial roles in cell growth, division, and differentiation. HER4 (ErbB4) is the least studied member of this family, partly because its expression is lower in later stages of development. Recent work has suggested that HER4 can play a role in metastasis by regulating cell migration and invasiveness; however, unlike EGFR and HER2, the precise role that HER4 plays in tumorigenesis is still unresolved. Early work on HER family proteins suggested that there are direct interactions between the four members, but to date, there has been no single study of all four receptors in the same cell line with the same biophysical method. Here, we quantitatively measure the degree of association between HER4 and the other HER family proteins in live cells with a time-resolved fluorescence technique called pulsed interleaved excitation fluorescence cross-correlation spectroscopy (PIE-FCCS). PIE-FCCS is sensitive to the oligomerization state of membrane proteins in live cells, while simultaneously measuring single-cell protein expression levels and diffusion coefficients. Our PIE-FCCS results demonstrate that HER4 interacts directly with all HER family members in the cell plasma membrane. The interaction between HER4 and other HER family members intensified in the presence of a HER4-specific ligand. Our work suggests that HER4 is a preferred dimerization partner for all HER family proteins, even in the absence of ligands.