Square-free Research Articles

Alternating state complexity of the set of primes and squarefree integers

AbstractWe show that the set of prime numbers has exponential alternating complexity, proving a conjecture by Fijalkow. We further show that the set of squarefree integers has essentially maximal possible alternating complexity.

Codes over $$\mathbb {F}_4$$ and $$\mathbb {F}_2 \times \mathbb {F}_2$$ and theta series of the corresponding lattices in quadratic fields

AbstractUsing codes defined over $$\mathbb {F}_4$$ F 4 and $$\mathbb {F}_2 \times \mathbb {F}_2$$ F 2 × F 2 , we simultaneously define the theta series of corresponding lattices for both real and imaginary quadratic fields $$\mathbb {Q}(\sqrt{d})$$ Q ( d ) with $$d \equiv 1\mod 4$$ d ≡ 1 mod 4 a square-free integer. For such a code, we use its weight enumerator to prove which term in the code’s corresponding theta series is the first to depend on the choice of d. For a given choice of real or imaginary quadratic field, we find conditions on the length of the code relative to the choice of quadratic field. When these conditions are satisfied, the generated theta series is unique to the code’s symmetric weight enumerator. We show that whilst these conditions ensure all non-equivalent codes will produce distinct theta series, for other codes that do not satisfy this condition, the length of the code and choice of quadratic field is not always enough to determine if the corresponding theta series will be unique.

On monogenity of certain number fields defined by x 2·3 s ·q r − m

Let K = Q ( α ) , where α is a root of the monic irreducible polynomial x 2 · 3 s · q r − m with m ≠ ± 1 is a square-free integer, r and s are two positive integers, and q is a prime of the form 3 k + 2 . In this article, we study the monogenity of the number field K and establish conditions on m for K to be monogenic. Further, we provide sufficient conditions on r, s and m for K to be not monogenic. Finally, we illustrate our results with examples.

Ranks of quadratic twists of Jacobians of generalized Mordell curves

Consider a two-parameter family of hyperelliptic curves Cq,b:y2=xq−bq defined over Q, and their Jacobians Jq,b where q is an odd prime and without loss of generality b is a non-zero squarefree integer. The curve Cq,b is a quadratic twist by b of Cq,1 (a generalized Mordell curve of degree q). First, we obtain a few upper bounds for the ranks e.g., if the class number of Q(ζq) is odd, q≡1(mod4) and any prime divisor of 2b not equal to q is a primitive root modulo q then rankJq,b(Q)≤(q−1)/2. Then we focus on q=5 and get the best possible bound (by 1) or even the exact value of rank (0). In particular, we found infinitely many b with any number of prime factors such that rankJ5,b(Q)=0. We deduce as conclusions the complete list (or the bounds for the number) of rational points on C5,b in such cases. Finally, we found for any given q infinitely many non-isomorphic curves Cq,b such that rankJq,b(Q)≥1.

Mertens's formula, k-full numbers and sum of two squares

In this article we prove generalizations of the well-known Mertens's formula. We also define some arithmetic functions related to the exponents of the primes in the prime factorization of the positive integers and study its average. Finally, we study square-free numbers, s-full numbers and perfect powers not exceeding x which can be represented as sums of two squares.

The average number of integral points on the congruent number curves

We show that the total number of non-torsion integral points on the elliptic curves ED:y2=x3−D2x, where D ranges over positive squarefree integers less than N, is O(N(log⁡N)−14+ϵ). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the 2-Selmer groups of the curves in this family.

Multivariate to bivariate reduction for noncommutative polynomial factorization

Based on Bergman's theorem, we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely,1.Given an n-variate noncommutative polynomial f∈F〈X〉 over a field F as an arithmetic circuit, computing a complete factorization of f into irreducible factors is deterministic polynomial-time reducible to factorization of a noncommutative bivariate polynomial g∈F〈x,y〉; the reduction transforms f into a circuit for g, and given a complete factorization of g, the reduction recovers a complete factorization of f in polynomial time.The reduction works both in the white-box and the black-box setting.2.We show over the field of rationals that bivariate linear matrix factorization problem for 4×4 matrices is at least as hard as factoring square-free integers and for 3×3 matrices it is in polynomial time.

Irreducible Z + -modules over a class of domains

Let d 1 , … , d r be pairwise relatively prime positive square-free integers, with d j ≥ 2 for all 1 ≤ j ≤ r . By using the elementary matrix methods, we give a complete classification of the irreducible Z + -modules over the domain Z [ d 1 , … , d r ] . Moreover, all of these irreducible Z + -modules are explicitly constructed.

Characterizing Finite Groups through Equitable Graphs: A Graph-Theoretic Approach

This paper introduces equitable graphs of Type I associated with finite groups. We investigate the connectedness and some graph-theoretic properties of these graphs for various groups. Furthermore, we establish the novel concepts of the equitable square-free number and the equitable group. Our study includes an analysis of the equitable graphs for specific equitable groups. Additionally, we determine the first, second and forgotten Zagreb topological indices for the equitable graphs of Type I constructed from certain groups. Finally, we derive the adjacency matrix for this graph type built from cyclic p-groups.

Distance between consecutive elements of the multiplicative group of integers modulo n

For a prime number $p$, we consider its primorial $P:=p\#$ and $U(P):={\left(\ZZ{P}\right)}^\times$ the set of elements of the multiplicative group of integers modulo $P$ which we represent as points anticlockwise on a circle of perimeter $P$. These points considered with wrap around modulo $P$ are those not marked by the Eratosthenes sieve algorithm applied to all primes less than or equal to $p$. In this paper, we are mostly concerned with providing formulas to count the number of gaps of a given even length $D$ in $U(P)$ which we note $K(D,P)$. This work, presented with different notations is closely related to [5]. We prove the formulas in three steps. Although only the last step relates to the problem of gaps in the Eratosthenes sieve (see Section 3.2.2) the previous formulas may be of interest to study occurrences of defined gaps sequences. <ul> <li>For a positive integer $n$, we prove a general formula based on the inclusion-exclusion principle to count the number of occurrences of configurations<sup>1</sup> in any subset of $\ZZ{n}$. (see Equation (7) in Theorem 2.1).</li> <li>For a square-free integer $P$, we particularize this formula when the subset of interest is $U(P)$. (see Equation (11) in Theorem 3.2).</li> <li>For a prime $p$ and its primorial $P:=p\#$, we particularize the formula again to study gaps in $U(P)$. Given a positive integer $D$ representing a distance on the circle, we give formulas to count $K(D,P)$ the number of gaps of length $D$ between elements of $U(P)$ (see Equation (15) and Section 4.1).</li> </ul> In addition, we provide a formula (see Equation (27) in Theorem 5.1) to count the number of occurrences of gaps of an even length $N$ that contain exactly $i$ elements of $U(P)$. <sup>1</sup> <small>A defined sequence of gaps between the elements of the subset; this is referred to as a constellation in [5].</small>

Square-free numbers in the intersection of Lehmer set and Piatetski-Shapiro sequence

<p>Let $ q $ be a sufficiently large odd integer, and let $ c \in\left(1, \frac{4}{3}\right) $. We denote $ R(c; q) $ as the count of square-free numbers in the intersection of the Lehmer set and the Piatetski-Shapiro sequence. By employing additive character properties to transform congruence equations and applying Kloosterman sums and methods of exponential sums, we derive a sharp asymptotic formula as $ q $ approaches infinity, which is significant for understanding the distribution properties of the Lehmer problem.</p>

On some sums involving the integral part function

Denote by [Formula: see text], [Formula: see text] and [Formula: see text] the number of representations of [Formula: see text] as a product of [Formula: see text] natural numbers, the number of distinct prime factors of [Formula: see text] and the characteristic function of the square-free integers, respectively. Let [Formula: see text] be the integral part of real number [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] for [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is an arbitrarily small positive number. These improve the corresponding results of Bordellès.

On moments of gaps between consecutive square-free numbers

On moments of gaps between consecutive square-free numbers

On the sum of a prime and a square-free number with divisibility conditions

Every integer greater than two can be expressed as the sum of a prime and a square-free number. Expanding on recent work, we provide explicit and asymptotic results when divisibility conditions are imposed on the square-free number. For example, we show for odd k≤105 and even k≤2⋅105 that any even integer n≥40 can be expressed as the sum of a prime and a square-free number coprime to k. We also discuss applications to other Goldbach-like problems.

Perfect codes in two-dimensional algebraic lattices

In the present paper, we investigate the existence of lattice perfect codes in two families of two-dimensional lattices built from the Minkowski embedding of the ring of integers of a quadratic field K=Q[l], where l is a square-free integer. We analyzed the forms and quantities of perfect codes for two of these families lattices, one Orthogonal and the other Non-Orthogonal and we verified that we always managed to find a perfect code in these two families of lattices. The greater the value of l the more perfect codes we get.

Counting square-free integers represented by binary quadratic forms of a fixed discriminant

Counting square-free integers represented by binary quadratic forms of a fixed discriminant

Explicit bounds for large gaps between squarefree integers

We obtain explicit forms of the current best known asymptotic upper bounds for gaps between squarefree integers. In particular we show, for any x≥2, that every interval of the form (x,x+11x1/5log⁡x] contains a squarefree integer. The constant 11 can be improved further, if x is assumed to be larger than a (very) large constant.

Sign changes of coefficients of half-integral weight Hecke eigenforms

Let f \mathfrak {f} be a cusp form of half-weight k + 1 / 2 k+1/2 and at most quadratic nebentype character whose Fourier coefficients are denoted as a f ( n ) \mathfrak {a}_\mathfrak {f}(n) . We study sign changes of the family { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} where t t is a square-free number and p p runs through the prime numbers. By taking use of the relationship between the cusp form f \mathfrak {f} of half-integral weight and the Shimura lift f t f_t , we establish that under certain conditions, the Hecke eigenforms of half-integral weight could be determined by sign changes of the sequence { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} .

The 2-Class Group of Certain Families of Imaginary Triquadratic Fields

ABSTRACT Our goal in this paper is to compute the 2-rank of the class group of the fields K = ℚ ( − 2 , q , d ) \[\mathbb{K}=\mathbb{Q}\left( \sqrt{-2},\sqrt{q},\sqrt{d} \right)\] for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2, 2), (2, 4) or (2, 2, 2).

The Dirichlet divisor problem over square-free integers and unitary convolutions

We obtain an asymptotic formula for the sum $\tilde{D}_2$ of the divisors of all square-free integers less than or equal to $x$, with error term $O(x^{1/2 + \epsilon})$. This improves the error term $O(x^{3/4 + \epsilon})$ presented in [7] obtained via analytical methods. Our approach is elementary and it is based on the connections between the function $\tilde{D}_2$ and unitary convolutions.