Square-free Research Articles

On a class of Lebesgue-Ljunggren-Nagell type equations

TextGiven odd, coprime integers a, b (a>0), we consider the Diophantine equation ax2+b2l=4yn, x,y∈Z, l∈N, n odd prime, gcd⁡(x,y)=1. We completely solve the above Diophantine equation for a∈{7,11,19,43,67,163}, and b a power of an odd prime, under the conditions 2n−1bl≢±1(mod a) and gcd⁡(n,b)=1. For other square-free integers a>3 and b a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers x, y with (gcd⁡(x,y)=1), l∈N and all odd primes n>3, satisfying 2n−1bl≢±1(mod a), gcd⁡(n,b)=1, and gcd⁡(n,h(−a))=1, where h(−a) denotes the class number of the imaginary quadratic field Q(−a). VideoFor a video summary of this paper, please visit https://youtu.be/Q0peJ2GmqeM.

Lower Bound for the Class Number of ℚn2+4

In this paper, we give an explicit lower bound for the class number of real quadratic field ℚd, where d=n2+4 is a square-free integer, using ωn which is the number of odd prime divisors of n.

On smooth square‐free numbers in arithmetic progressions

A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact one can find such $s$ with $s = p^{O(1)}$. Using bounds on double Kloosterman sums due to M. Z. Garaev (2010) we prove this conjecture in a stronger form $s \le p^{3/2 + o(1)}$ and also consider more general versions of this question replacing $p$-smoothness of $s$ by the stronger condition of $p^{\alpha}$-smoothness. Using bounds on multiplicative character sums and a sieve method, we also show that we can represent all residue classes by a positive square-free integer $s\le p^{2+o(1)}$ which is $p^{1/(4e^{ /2})+o(1)}$-smooth. Additionally, we obtain stronger results for almost all primes $p$.

Maximal equicontinuous generic factors and weak model sets

The orbit closures of regular model sets generated from a cut-and-project scheme given by a co-compact lattice \begin{document}$ {\mathcal L}\subset G\times H $\end{document} and compact and aperiodic window \begin{document}$ W\subseteq H $\end{document} , have the maximal equicontinuous factor (MEF) \begin{document}$ (G\times H)/ {\mathcal L} $\end{document} , if the window is toplogically regular. This picture breaks down, when the window has empty interior, in which case the MEF is trivial, although \begin{document}$ (G\times H)/ {\mathcal L} $\end{document} continues to be the Kronecker factor for the Mirsky measure. As this happens for many interesting examples like the square-free numbers or the visible lattice points, a weaker concept of topological factors is needed, like that of generic factors [ 24 ]. For topological dynamical systems that possess a finite invariant measure with full support ( \begin{document}$ E $\end{document} -systems) we prove the existence of a maximal equicontinuous generic factor (MEGF) and characterize it in terms of the regional proximal relation. This part of the paper profits strongly from previous work by McMahon [ 33 ] and Auslander [ 2 ]. In Sections 3 and 4 we determine the MEGF of orbit closures of weak model sets and use this result for an alternative proof (of a generalization) of the fact [ 34 ] that the centralizer of any \begin{document}$ {\mathcal B} $\end{document} -free dynamical system of Erdős type is trivial.

Twists of hyperelliptic curves by integers in progressions modulo $p$

Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral point.

Indecomposable integers in real quadratic fields

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q(D) where D>1 is a squarefree integer. Their conjecture was later disproved by Kala for D≡2mod4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D≡1,2,3mod4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O(D).

Shimura and Shintani liftings of certain cusp forms of half-integral and integral weights

In [4], W. Kohnen constructed explicit Shintani lifts from the space of cusp forms of weight $2k$ on $Γ_0(N)$, which are mapped into the plus space $S^+_{k+1/2}(Γ_0(4N))$, where $N$ is an odd integer. This lifting is adjoint to the (modified) Shimura map defined by him with respect to the Petersson scalar product. Using this construction along with the theory of newforms on $S^+_{k+1/2}(Γ_0(4N))$ (where $N$ is an odd square-free natural number) developed in [3], he derived explicit Waldspurger theorem for the newforms belonging to the space $S^+_{k+1/2}(Γ_0(4N))$. Further in this work, Kohnen considered the $\pm 1$ eigen subspaces corresponding to the Atkin-Lehner involution and showed that the intersection of this $\pm$ spaces with the corresponding newform spaces are isomorphic under the Shimura correspondence. A natural question is whether a similar result can be obtained for the intersection of this $(\pm1)$ subspaces with the space of oldforms. In this direction, S. Choi and C. H. Kim [2] considered the case where $N$ is an odd prime $p$ and constructed similar Shimura and Shintani maps between subspaces of forms of half-integral and integral weights. The subspaces considered by Choi and Kim are nothing but $+1$ eigenspace under the Fricke involution. In this paper, we generalise the work of Choi and Kim to the case where $N$ is an odd square-free natural number.

SMOOTH SQUAREFREE AND SQUAREFULL INTEGERS IN ARITHMETIC PROGRESSIONS

We obtain new lower bounds on the number of smooth squarefree integers up to x in residue classes modulo a prime p, relatively large compared to x, which in some ranges of p and x improve that of Balog and Pomerance. We also obtain an estimate on the smallest squarefull number in almost all residue classes modulo a prime p.

Приближение квадратичных алгебраических решёток целочисленными решётками

Данная работа посвящена вопросам приближения квадратичной алгебраической решётки целочисленной решёткой. В ней вычисляются расстояния между квадратичной алгебраической решёткой и целочисленной решёткой, когда они заданы числителем и знаменателем подходящей дроби к корню квадратному из простого числа ???? вида ???? = 2 или ???? = 4???? + 3.Результаты данной работы позволяют изучать вопросы о наилучших приближенияхквадратичных алгебраических решёток целочисленными решётками.

English

Let $f=\sum_{n=1}^{\infty}a(n)q^{n}\in S_{k+1/2}(N,\chi_{0})$ be a non-zero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus $\chi_{0}$ where the Fourier coefficients $a(n)$ are real. Bruinier and Kohnen conjectured that the signs of $a(n)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies $\{a(t n^{2})\}_{n}$ where $t$ is a squarefree integer such that $a(t)\neq 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that $\{a(t n^{2})\}_{n}$ is equidistributed over any arithmetic progression $n\equiv d\text{ mod }q$.

Rational Points of Some Elliptic Curves Related to the Tilings of the Equilateral Triangle

Let n be a positive and squarefree integer. We show that the equilateral triangle can be dissected into ncdot k^2 congruent triangles for some k if and only if nle 3, or at least one of the curves C_n :y^2 =x(x-n)(x+3n) and C_{-n} : y^2 =x(x+n)(x-3n) has a rational point with yne 0. We prove that if p is a positive prime such that pequiv 7 (mod 24), then C_p and C_{-p} do not have such points. Consequently, for these primes the equilateral triangle cannot be dissected into pcdot k^2 congruent triangles for any k.

The morphology of

What are the units, irreducibles, and primes of the ring ${\mathbb{Z}}\sqrt n $, the set of all numbers $a + b\sqrt n $ where a and b are integers and n is a fixed positive square-free integer? In the ring ${\mathbb{Z}}$, primes and irreducibles are synonymous and its units are ±1. ${\mathbb{Z}}\sqrt n $ is wilder, and our modest goal here is to catalogue all such numbers for ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $, where a and b range from 0 to 10; the result appears in Figure 1. Here are a few teasers that may induce a reader to read on: $3 + \sqrt {10} $ is a unit; 2, 3, 5, and 7 are irreducibles, but not 31; and 7 is the least positive integer that is prime in both ${\mathbb{Z}}$ and ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $.

A simple criterion for the class number of a quadratic number field to be one

Let [Formula: see text] be a square-free positive integer, [Formula: see text] if [Formula: see text] and [Formula: see text] otherwise. Let [Formula: see text] and [Formula: see text] be integers, where [Formula: see text] is a prime. Suppose that [Formula: see text] for some integer [Formula: see text]. Suppose that there exist integers [Formula: see text] and [Formula: see text] such that [Formula: see text]. We prove that if [Formula: see text] is [Formula: see text] or a prime for all integers [Formula: see text] with [Formula: see text], then the class number of the field [Formula: see text] is [Formula: see text].

Biased behavior of weighted Mertens sums

Using convexity properties of reciprocals of zeta functions, especially the reciprocal of the Riemann zeta function, we show that certain weighted Mertens sums are biased in favor of square-free integers with an odd number of prime factors. We study such type of bias for different ranges of the parameters and then consider generalizations to Mertens sums supported on semigroups of integers generated by relatively large subsets of prime numbers. We further obtain a wider range for the parameters both unconditionally and then conditionally on the Riemann Hypothesis. At the same time, we extend to certain semigroups, two classical summation formulas originating from the works of Landau concerning the behavior of derivatives of the reciprocal of the Riemann zeta function at [Formula: see text].

Representation of Algebraic Integers as Sum of Units over the Real Quadratic Fields

In this paper we generalize Jacobsons results by proving that any integer in is a square-free integer), belong to . All units of are generated by the fundamental unit having the forms
 our generalization build on using the conditions
 This leads us to classify the real quadratic fields into the sets Jacobsons results shows that and Sliwa confirm that and are the only real quadratic fields in .

Fields of values of odd-degree irreducible characters

In this paper we clarify the quadratic irrationalities that can be admitted by an odd-degree complex irreducible character χ of an arbitrary finite group. Write Q(χ) to denote the field generated over the rational numbers by the values of χ, and let d>1 be a square-free integer. We prove that if Q(χ)=Q(d) then d≡1 (mod 4) and if Q(χ)=Q(−d), then d≡3 (mod 4). This follows from the main result of this paper: either i∈Q(χ) or Q(χ)⊆Q(exp⁡(2πi/m)) for some odd integer m≥1.

Ternary quadratic forms and the class numbers of imaginary quadratic fields

We establish a series of relations between the numbers of solutions of some positive definite ternary quadratic forms and the class numbers of the corresponding imaginary quadratic fields. In particular, we obtain an explicit formula for the ternary quadratic form , where p and q are odd primes. Let h(d) be the class number of , and for a ternary quadratic form Q, we use for the number of integral solutions (x, y, z) of the equation . We give two examples to illustrate our results: (i) For , if m is a square-free positive integer with , then(ii) for , if m is a square-free positive integer with , then

A Verified Implementation of the Berlekamp\u2013Zassenhaus Factorization Algorithm

We formally verify the Berlekamp–Zassenhaus algorithm for factoring square-free integer polynomials in Isabelle/HOL. We further adapt an existing formalization of Yun’s square-free factorization algorithm to integer polynomials, and thus provide an efficient and certified factorization algorithm for arbitrary univariate polynomials. The algorithm first performs factorization in the prime field mathrm {GF}(p){} and then performs computations in the ring of integers modulo p^k, where both p and k are determined at runtime. Since a natural modeling of these structures via dependent types is not possible in Isabelle/HOL, we formalize the whole algorithm using locales and local type definitions. Through experiments we verify that our algorithm factors polynomials of degree up to 500 within seconds.

Associations between personality traits, substance use, and depression in Romanian medical students

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q(D) where D>1 is a squarefree integer. Their conjecture was later disproved by Kala for D≡2mod4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D≡1,2,3mod4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O(D).

The first negative eigenvalue of Yoshida lifts

We prove that for any given $$\epsilon >0$$ , the first negative eigenvalue of the Yoshida lift F of a pair of elliptic cusp forms f, g having square-free levels (where g has weight 2 and satisfies $$(\log Q_{g})^2 \ll \log Q_f$$ ), occurs before $$c_{\epsilon } \cdot Q_F^{1/2-2 \theta + \epsilon }$$ ; where $$Q_F,Q_f,Q_g$$ are the analytic conductors of F, f, g respectively, $$\theta < 1/4$$ , and $$c_{\epsilon }$$ is a constant depending only on $$\epsilon $$ .