Square-free Research Articles

Injectivity of compressing maps on the set of primitive sequences modulo square-free odd integers

Let p1, p2,?,pr be distinct odd primes and m = p1p2?pr. Let f(x) be a primitive polynomial of degree n over ?/m?$\mathbb {Z}/m\mathbb {Z}$. Denote by L(f) the set of primitive linear recurring sequences generated by f(x). A map ? on ?/m?$\mathbb {Z}/m\mathbb {Z}$ naturally induces a map ??$\widehat {\psi }$ on L(f), mapping a sequence (?,s?(i?1),s?(i),s?(i+1),?)$(\dots ,\underline {s}({i-1}),\underline {s}(i),\underline {s}({i+1}),\dots )$ to (?,?(s?(i?1)),?(s?(i)),?(s?(i+1)),?)$(\dots ,\psi (\underline {s}({i-1})),\psi (\underline {s}(i)),\psi (\underline {s}({i+1})),\dots )$. Previous results gave sufficient conditions under which modular functions induce injective maps on L(f). In this article we give an inequality which holds for large enough n. If this inequality holds, then the injectivity of ??$\hat {\psi }$ is clearly determined for any map ? on ?/m?$\mathbb {Z}/m\mathbb {Z}$. Particularly, the modular function ?(a)=a mod M induces an injective map on L(f) for any M?2≤i??:i?m$M\in \left \{{2\leq i\in \mathbb {Z}:i \nmid m}\right \}$.

An Algorithmic Construction of Finite Elliptic Curves of Order Divisible by a Large Prime

Given a square-free integer Δ < 0, we present an algorithm constructing a pair of primes p and q such that q|p + 1-t and 4p-t2 = Δf2, where |t| ≤ 2√p for some integers f, t. Together with a CM method presented in the paper, such primes p and q are used for a construction of an elliptic curve E over a finite field $\mathbb{F}_p$ such that the order of E is divisible by a large prime. It is shown that our algorithm works in polynomial time.

Bell’s Ternary Quadratic Forms and Tunnel’s Congruent Number Criterion Revisited

Bell’s theorem determines the number of representations of a positive integer in terms of the ternary quadratic forms x2+by2+cz2 with b,c {1,2,4,8}. This number depends only on the number of representations of an integer as a sum of three squares. We present a modern elementary proof of Bell’s theorem that is based on three standard Ramanujan theta function identities and a set of five so-called three-square identities by Hurwitz. We use Bell’s theorem and a slight extension of it to find explicit and finite computable expressions for Tunnel’s congruent number criterion. It is known that this criterion settles the congruent number problem under the weak Birch-Swinnerton-Dyer conjecture. Moreover, we present for the first time an unconditional proof that a square-free number n 3(mod 8) is not congruent.

On Bourgain's bound for short exponential sums and squarefree numbers

We use Bourgain's recent bound for short exponential sums to prove certain independence results related to the distribution of squarefree numbers in arithmetic progressions.

On the classification of almost null rings

An almost null ring is a ring $R$ in which for all $a,b\in R$, $a^3=0$, $Ma^2=0$ for some square-free integer $M$ that depends on $a$ and $ab= ka^{2}=l b^{2}$ for some integers $k,l$. This paper is devoted to the classification of the almost null rings.

Ergodic Properties of Visible Lattice Points

Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as ℬ-free numbers with Sarnak’s conjecture on the “randomness” of the Mobius function; another is the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.

Ergodic properties of visible lattice points

Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as ℬ-free numbers with Sarnak’s conjecture on the “randomness” of the Möbius function; another is the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.

Phénomène de Moser-Newman pour les nombres sans facteur carré

Let μ denotes the Mobius function and t = (t(n))n∈N be the Thue-Morse sequence, defined by t(n) = +1 if the number of 1 in the dyadic representation of n is even and t(n) = −1 otherwise. The aim of this work is to give an asymptotic formula for S(N) = Σn<N μ2(n)t(n) and to prove that S(N) < 0 for N big enough. This shows that square-free integers provide a first example of non-linear Moser-Newman phenomenon. Our method gives a similar result for z-th power free integers.

GCD sums and complete sets of square-free numbers

It is proved that \[ \sum_{k,{\ell}=1}^N\frac{\gcd(n_k,n_{\ell})}{\sqrt{n_k n_{\ell}}} \ll N\exp\left(C\sqrt{\frac{\log N \log\log\log N}{\log\log N}}\right) \] holds for arbitrary integers $1\le n_1<\cdots < n_N$. This bound is essentially better than that found in a recent paper of Aistleitner, Berkes, and Seip and can not be improved by more than possibly a power of $1/\sqrt{\log\log\log N}$. The proof relies on ideas from classical work of G\'{a}l, the method of Aistleitner, Berkes, and Seip, and a certain completeness property of extremal sets of square-free numbers.

Power integral bases for certain pure sextic fields

In this paper, we characterize whether the pure sextic fields [Formula: see text] with square-free integers m ≢ ±1( Mod 9) have power integral bases or do not; if m ≡ 2, 3 (Mod 4), then [Formula: see text] have power integral bases. We prove this by determining relative integral bases of such fields with respect to their cubic and quadratic subfields. Based on the works of Kovács and Pethő, several examples on application of monogenic fields to CNS (Canonical Number System) are shown.

A Dynamical Point of View on the Set ofℬ-Free Integers

Sarnak has recently initiated the study of the Möbius function and its square, the characteristic function of square-free integers, from a dynamical point of view, introducing the Möbius flow and the square-free flow as the action of the shift map on the respective subshfits generated by these functions. In this paper, we extend the study of the square-free flow to the more general context of |$\mathscr {B}$|-free integers, that is to say integers with no factor in a given family |$\mathscr {B}$| of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, we prove in particular that the distribution of patterns in the characteristic function of the |$\mathscr {B}$|-free integers follows a shift-invariant probability measure, and gives rise to a measurable dynamical system isomorphic to a specific minimal rotation on a compact group. As a by-product, we obtain the abundance of twin |$\mathscr {B}$|-free numbers. Moreover, we show that the distribution of patterns in small intervals of the form |$[N,N+ \sqrt {N})$| also conforms to the same measure. When elements of |$\mathscr {B}$| are squares, we introduce a generalization of the Möbius function, and discuss a conjecture of Chowla in this broader context.

A NOTE ON TERNARY CYCLOTOMIC POLYNOMIALS

Let <TEX>${\Phi}_n(x)={\sum}^{{\phi}(n)}_{k=0}a(n,k)x^k$</TEX> denote the n-th cyclotomic polynomial. In this note, let p < q < r be odd primes, where <TEX>$q{\not{\equiv}}1$</TEX> (mod p) and <TEX>$r{\equiv}-2$</TEX> (mod pq), we construct an explicit k such that a(pqr, k) = -2.

On the Cayley Isomorphism Problem for Cayley Objects of Nilpotent Groups of Some Orders

We give a necessary condition to reduce the Cayley isomorphism problem for Cayley objects of a nilpotent or abelian group $G$ whose order satisfies certain arithmetic properties to the Cayley isomorphism problem of Cayley objects of the Sylow subgroups of $G$ in the case of nilpotent groups, and in the case of abelian groups to certain natural subgroups. As an application of this result, we show that ${\mathbb Z}_q\times{\mathbb Z}_p^2\times{\mathbb Z}_m$ is a CI-group with respect to digraphs, where $q$ and $p$ are primes with $p^2 &lt; q$ and $m$ is a square-free integer satisfying certain arithmetic conditions (but there are no other restrictions on $q$ and $p$).

Hilbert genus fields of biquadratic fields

The Hilbert genus field of the real biquadratic field \(K = \mathbb{Q}\left( {\sqrt \delta ,\sqrt d } \right)\) is described by Yue (2010) and Bae and Yue (2011) explicitly in the case δ = 2 or p with p ≡ 1 mod 4 a prime and d a squarefree positive integer. In this article, we describe explicitly the Hilbert genus field of the imaginary biquadratic field \(K = \mathbb{Q}\left( {\sqrt \delta ,\sqrt d } \right)\), where δ = −1,−2 or −p with p ≡ 3mod 4 a prime and d any squarefree integer. This completes the explicit construction of the Hilbert genus field of any biquadratic field which contains an imaginary quadratic subfield of odd class number.

The group of squarefree integers

We investigate the consequences of the elementary observation that the squarefree numbers form a group under the operation lcmgcd. In particular, we discuss the characters on this group, one of which is the Möbius function, as well as the finite subgroups D(k) formed from the divisors of a given squarefree integer k. We show further how a convolution, naturally based on this operation, leads to the factorization of various arithmetical matrices and the evaluation of the eigenvalues. We discuss briefly the associated L-functions. Finally, we generalize the operation to other subsets of N.

Integers with a given number of divisors

TextFor any positive integer n, let n=q1⋯qs be the prime factorization of n with q1≥⋯≥qs>1. A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is p1q1−1⋯psqs−1, where pk denotes the kth prime. Let ⌊x⌋ be the largest integer not exceeding x. In 2006, Brown proved that all square-free integers are ordinary and the set of all ordinary integers has asymptotic density one. In this paper, we prove that, if q⌊s⌋≥9(logs)2, then n is ordinary. Furthermore, the set of such integers n has asymptotic density one. We also determine all ordinary integers which are not divisible by any fifth power of a prime. VideoFor a video summary of this paper, please visit http://youtu.be/UeIMWjRFUnA.

On the distribution of eigenvalues of Maass forms on certain moonshine groups

In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups Γ 0 ( N ) + \Gamma _0(N)^+ , where N N is a positive, square-free integer. After we prove that Γ 0 ( N ) + \Gamma _0(N)^+ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an “average” Weyl’s law for the distribution of eigenvalues of Maass forms, from which we prove the “classical” Weyl’s law as a special case. The groups corresponding to N = 5 N=5 and N = 6 N=6 have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for Γ 0 ( 5 ) + \Gamma _0(5)^+ than for Γ 0 ( 6 ) + \Gamma _0(6)^+ . We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl’s laws. In addition, we employ Hejhal’s algorithm, together with recently developed refinements from [H. Then, Computing large sets of consecutive Maass forms, in preparation], and numerically determine the first 3557 3557 eigenvalues of Γ 0 ( 5 ) + \Gamma _0(5)^+ and the first 12474 12474 eigenvalues of Γ 0 ( 6 ) + \Gamma _0(6)^+ . With this information, we empirically verify some conjectured distributional properties of the eigenvalues.

Bounded Gaps between Products of Special Primes

In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for any r ≥ 2, all of which are in a given set of primes. His results yield applications to the divisibility of class numbers and the triviality of ranks of elliptic curves. In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. We then revisit Thorne’s applications and give a better bound in each case.

Systems of imprimitivity for the clifford group

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of k-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).

Dynamical Properties of k-Free Lattice Points

We revisit the visible points of a lattice in Euclidean $n$-space together with their generalisations, the $k$th-power-free points of a lattice, and study the corresponding dynamical system that arises via the closure of the lattice translation orbit. Our analysis extends previous results obtained by Sarnak and by Cellarosi and Sinai for the special case of square-free integers and sheds new light on previous joint work with Peter Pleasants.