Square-free Research Articles

On power integral bases for certain pure sextic fields

In their paper [1], Shahzad Ahmad et al. given a characterization on any pure sextic number field Q(m1/6) with square-free integers m satisfying m 6 ±1 (mod 9) to have a power integral bases or do not. In this paper, for these results, we give a new easier proof than that given in [1]. We further investigate the cases m 1 (mod 4) independently to the satisfaction of m2 1 (mod 9), m 1 (mod 9), and the number fields defined by x2r3t−m, where r, t are two non-negative integers, and m is a square free integer are investigated. The proposed proofs are based on Dedekind’s criterion and on prime ideal factorization.

Refinement of the upper bound of the radius of a circular integral points set

Let n be a product of the prime numbers whose positive integer powers are of the form a2+Db2 where D> 4 is a square-free number and a, b are positive integers. For n≤ 3072, we obtained a refinement of the upper bound of the radius of a circular points set which was previously given in Tables 1 and 2 of Ganbileg's paper. In order to prove this, we showed that there are points on the circle with the radius R=n √D/2D such that mutual distances between these points are all integers. Consequently, if n is a product of the prime numbers whose squares are of the form a2+Db2, then we showed that there are τ(n) points on the circle with the radius R=n √D/2D such that mutual distances between these points are all integer numbers, where τ(n) is the number of all positive divisors of n.

Elliptic curves with conductor having $n$ prime factors

We prove that for any integer $n \ge 2$, there are infinitely many elliptic curves over $\mathbb {Q}$ with a rational point of order 2 (resp. 3) whose conductor is a square-free integer having $n$ prime factors.

The 8-rank of the narrow class group and the negative Pell equation

Abstract Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let $\mathcal {D}$ denote the set of positive squarefree integers having no prime factors congruent to $3$ modulo $4$ . Stevenhagen [19] conjectured that the density of d in $\mathcal {D}$ such that the negative Pell equation $x^2-dy^2=-1$ is solvable with $x, y \in \mathbb {Z}$ is $58.1\%$ , to the nearest tenth of a percent. By studying the distribution of the $8$ -rank of narrow class groups $\operatorname {\mathrm {Cl}}^+(d)$ of $\mathbb {Q}(\sqrt {d})$ , we prove that the infimum of this density is at least $53.8\%$ .

Khintchine’s theorem with rationals coming from neighborhoods in different places

The Duffin–Schaeffer Conjecture answers a question on how well one can approximate irrationals by rational numbers in reduced form (an imposed condition) where the accuracy of the approximation depends on the rational number. It can be viewed as an analogue to Khintchine’s theorem with the added restriction of only allowing rationals in reduced form. Other conditions such as numerator or denominator a prime, a square-free integer, or an element of a particular arithmetic progression, etc. have also been imposed and analogues of Khintchine’s theorem studied. We prove versions of Khintchine’s theorem where the rational numbers are sourced from a ball in some completion of [Formula: see text] (i.e. Euclidean or [Formula: see text]-adic), while the approximations are carried out in a distinct second completion. Finally, by using a mass transference principle for Hausdorff measures, we are able to extend our results to their corresponding analogues with Haar measures replaced by Hausdorff measures, thereby establishing an analogue of Jarník’s theorem.

Proof of the Kakeya set conjecture over rings of integers modulo square-free N

A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least $C_{n,\epsilon} N^{n - \epsilon}$ for any $\epsilon$ -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime $N$. We also show that the case of general $N$ can be reduced to lower bounding the $\mathbb{F}_p$ rank of the incidence matrix of points and hyperplanes over $(\mathbb{Z}/p^k\mathbb{Z})^n$.

On the Diophantine Equation $$dx^2+p^{2a}q^{2b}=4y^p$$

We investigate the solvability of the Diophantine equation in the title, where \(d>1\) is a square-free integer, p, q are distinct odd primes and x, y, a, b are unknown positive integers with \(\gcd (x,y)=1\). We describe all the integer solutions of this equation, and then use the main finding to deduce some results concerning the integers solutions of some of its variants. The methods adopted here are elementary in nature and are primarily based on the existence of the primitive divisors of certain Lehmer numbers.

Randomness of Möbius coefficients and Brownian motion: growth of the Mertens function and the Riemann hypothesis

The validity of the Riemann hypothesis (RH) on the location of the non-trivial zeros of the Riemann ζ-function is directly related to the growth of the Mertens function , where μ(k) is the Möbius coefficient of the integer k; the RH is indeed true if the Mertens function goes asymptotically as M(x) ∼ x 1/2+ϵ , where ϵ is an arbitrary strictly positive quantity. We argue that this behavior can be established on the basis of a new probabilistic approach based on the global properties of the Mertens function, namely, based on reorganizing globally in distinct blocks the terms of its series. With this aim, we focus attention on the square-free numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of square-free numbers, the average number of prime divisors, the Erdős–Kac theorem for square-free numbers, etc. These results point to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk. We also present an argument in favor of the thesis that the validity of the RH also implies the validity of the generalized RH for the Dirichlet L-functions. Next we study the local properties of the Mertens function, i.e. its variation induced by each Möbius coefficient restricted to the square-free numbers. Motivated by the natural curiosity to see how closely to a purely random walk any sub-sequence is extracted by the sequence of the Möbius coefficients for the square-free numbers, we perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity; together with several frequency tests within a block, the list of our tests includes those for the longest run of ones in a block, the binary matrix rank test, the discrete Fourier transform test, the non-overlapping template matching test, the entropy test, the cumulative sum test, the random excursion tests, etc, for a total of 18 different tests. The successful outputs of all these tests (each of them with a level of confidence of 99% that all the sub-sequences analyzed are indeed random) can be seen as impressive ‘experimental’ confirmations of the Brownian nature of the restricted Möbius coefficients and the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however extremely improbable.

An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative

Let N N be one of the 38 38 distinct square-free integers such that the arithmetic group Γ 0 ( N ) + \Gamma _0(N)^+ has genus one. We constructed canonical generators x N x_N and y N y_N for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of x N x_N , which we express as a polynomial in y N y_N with coefficients that are rational functions in x N x_N . As a corollary, we prove that for any point e e in the upper half-plane which is fixed by an element of Γ 0 ( N ) + \Gamma _0(N)^+ , one can explicitly evaluate x N ( e ) x_N(e) and y N ( e ) y_N(e) . As it turns out, each value x N ( e ) x_N(e) and y N ( e ) y_N(e) is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of x N ( τ ) x_N(\tau ) and y N ( τ ) y_N(\tau ) at any CM point τ \tau , including elliptic points, for any genus one group Γ 0 ( N ) + \Gamma _0(N)^+ . Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity.

Unit group structure of the quotient ring of a quadratic ring

Let [Formula: see text] be the rational filed. For a square-free integer [Formula: see text] with [Formula: see text], we denote by [Formula: see text] the quadratic field. Let [Formula: see text] be the ring of algebraic integers of [Formula: see text]. In this paper, we completely determine the unit group of the quotient ring [Formula: see text] of [Formula: see text] for an arbitrary prime [Formula: see text] in [Formula: see text], where [Formula: see text] has the unique factorization property, and [Formula: see text] is a rational integer.

CONSECUTIVE SQUARE-FREE NUMBERS IN PIATETSKI-SHAPIRO SEQUENCES

AbstractUsing a method due to Rieger [‘Remark on a paper of Stux concerning squarefree numbers in non-linear sequences’, Pacific J. Math.78(1) (1978), 241–242], we prove that the Piatetski-Shapiro sequence defined by $\{\lfloor n^c \rfloor : n\in \mathbb {N}\}$ contains infinitely many consecutive square-free integers whenever $1<c<3/2$ .

A variance fork-free numbers in arithmetic progressions of given modulus

An asymptotic formula for the variance of squarefree numbers in arithmetic progressions of given modulus was obtained by Nunes (see reference [3]). We improve one of the error terms.

On Yokoi’s Invariants and the Ankeny–Artin–Chowla conjecture

Let [Formula: see text] be a positive square-free integer and [Formula: see text] be the fundamental unit of the real quadratic field [Formula: see text]. The Ankeny–Artin–Chowla (AAC) conjecture asserts that [Formula: see text] (mod [Formula: see text]) for primes [Formula: see text] (mod 4), which still remains unsolved. In this paper, sufficient conditions for [Formula: see text] have been given in terms of Yokoi’s invariants [Formula: see text] and [Formula: see text], and it has been shown that the AAC conjecture is true in some special cases.

Search Heuristics and Constructive Algorithms for Maximally Idempotent Integers

Previous work established the set of square-free integers n with at least one factorization n=p¯q¯ for which p¯ and q¯ are valid RSA keys, whether they are prime or composite. These integers are exactly those with the property λ(n)∣(p¯−1)(q¯−1), where λ is the Carmichael totient function. We refer to these integers as idempotent, because ∀a∈Zn,ak(p¯−1)(q¯−1)+1≡na for any positive integer k. This set was initially known to contain only the semiprimes, and later expanded to include some of the Carmichael numbers. Recent work by the author gave the explicit formulation for the set, showing that the set includes numbers that are neither semiprimes nor Carmichael numbers. Numbers in this last category had not been previously analyzed in the literature. While only the semiprimes have useful cryptographic properties, idempotent integers are deserving of study in their own right as they lie at the border of hard problems in number theory and computer science. Some idempotent integers, the maximally idempotent integers, have the property that all their factorizations are idempotent. We discuss their structure here, heuristics to assist in finding them, and algorithms from graph theory that can be used to construct examples of arbitrary size.

On Hilbert genus fields of imaginary cyclic quartic fields

Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let $\varepsilon_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. The main aim of this paper is to determine explicitly the Hilbert genus field of the imaginary cyclic quartic fields of the form $\mathbb{Q}(\sqrt{-a\varepsilon_p\sqrt{p}})$.

Simultaneous indivisibility of class numbers of pairs of real quadratic fields

For a square-free integer t, Byeon (Proc. Am. Math. Soc. 132:3137–3140, 2004) proved the existence of infinitely many pairs of quadratic fields $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{tD})$$ with $$D > 0$$ such that the class numbers of all of them are indivisible by 3. In the same spirit, we prove that for a given integer $$t \ge 1$$ with $$t \equiv 0 \pmod {4}$$ , a positive proportion of fundamental discriminants $$D > 0$$ exist for which the class numbers of both the real quadratic fields $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{D + t})$$ are indivisible by 3. This also addresses the complement of a weak form of a conjecture of Iizuka (J. Numb. Theory, 184:122–127, 2018). As an application of our main result, we obtain that for any integer $$t \ge 1$$ with $$t \equiv 0 \pmod {12}$$ , there are infinitely many pairs of real quadratic fields $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{D + t})$$ such that the Iwasawa $$\lambda $$ -invariants associated with the basic $$\mathbb {Z}_{3}$$ -extensions of both $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{D + t})$$ are 0. For $$p = 3$$ , this supports Greenberg’s conjecture which asserts that $$\lambda _{p}(K) = 0$$ for any prime number p and any totally real number field K.

Consecutive square-free numbers and square-free primitive roots

Let [Formula: see text] be a positive integer and let [Formula: see text] be an odd prime. For [Formula: see text], we study the distribution of consecutive square-free numbers of the forms [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text]. In addition, we study the distribution of consecutive square-free primitive roots modulo [Formula: see text] of the forms [Formula: see text],[Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], respectively.

105.20 Revisiting even and odd square-free numbers

105.20 Revisiting even and odd square-free numbers

English

We show that there exist infinitely many consecutive square-free numbers of the form n2 + 1, n2 + 2. We also establish an asymptotic formula for the number of such square-free pairs when n does not exceed given sufficiently large positive number.

An arithmetic criterion for graphs being determined by their generalized Aα-spectra

Let G be a graph on n vertices, its adjacency matrix and degree diagonal matrix are denoted by A(G) and D(G), respectively. In 2017, Nikiforov [20] introduced the matrix Aα(G)=αD(G)+(1−α)A(G) for α∈[0,1]. The Aα-spectrum of a graph G consists of all the eigenvalues (including the multiplicities) of Aα(G). A graph G is said to be determined by the generalized Aα-spectrum (or, DGAαS for short) if whenever H is a graph such that H and G share the same Aα-spectrum and so do their complements, then H is isomorphic to G. In this paper, when α is rational, we present a simple arithmetic condition for a graph being DGAαS. More precisely, put Acα:=cαAα(G), here cα is the smallest positive integer such that Acα is an integral matrix. Let W˜α(G)=[1,Acα1cα,…,Acαn−11cα], where 1 denotes the all-ones column vector. We prove that if det⁡W˜α(G)2⌊n2⌋ is an odd and square-free integer and the rank of W˜α(G) is full over Fp for each odd prime divisor p of cα, then G is DGAαS except for even n and odd cα(⩾3). By our obtained results in this paper we may deduce the main results in [24] and [27].