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Artykuły w czasopismach na temat „K free integers”

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Data publikacji: 29 marca 2025

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1

Wu, Xia, i Yan Qin. "Rational Points of Elliptic Curve y2=x3+k3". Algebra Colloquium 25, nr 01 (22.01.2018): 133–38. http://dx.doi.org/10.1142/s1005386718000081.

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Let E be an elliptic curve defined over the field of rational numbers ℚ. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed(ℚ) is zero. Let E(k) denote the elliptic curve y2 = x3 + k. Then the quadratic twist E(1)d of E(1) by d is the elliptic curve [Formula: see text]. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d ≡ r (mod 24) such that rank [Formula: see text], using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E(k3)(ℚ) has rank zero.

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2

Haukkanen, Pentti. "Arithmetical functions associated with conjugate pairs of sets under regular convolutions". Notes on Number Theory and Discrete Mathematics 28, nr 4 (24.10.2022): 656–65. http://dx.doi.org/10.7546/nntdm.2022.28.4.656-665.

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Two subsets P and Q of the set of positive integers is said to form a conjugate pair if each positive integer n possesses a unique factorization of the form n = ab, a ∈ P, b ∈ Q. In this paper we generalize conjugate pairs of sets to the setting of regular convolutions and study associated arithmetical functions. Particular attention is paid to arithmetical functions associated with k-free integers and k-th powers under regular convolution.

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3

Minh, Nguyen Quang. "A Generalisation of Maximal (k,b)-Linear-Free Sets of Integers". Journal of Combinatorial Mathematics and Combinatorial Computing 120, nr 1 (30.06.2024): 315–21. http://dx.doi.org/10.61091/jcmcc120-28.

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Fix integers k , b , q with k ≥ 2 , b ≥ 0 , q ≥ 2 . Define the function p to be: p ( x ) = k x + b . We call a set S of integers \emph{ ( k , b , q ) -linear-free} if x ∈ S implies p i ( x ) ∉ S for all i = 1 , 2 , … , q − 1 , where p i ( x ) = p ( p i − 1 ( x ) ) and p 0 ( x ) = x . Such a set S is maximal in [ n ] := { 1 , 2 , … , n } if S ∪ { t } , ∀ t ∈ [ n ] ∖ S is not ( k , b , q ) -linear-free. Let M k , b , q ( n ) be the set of all maximal ( k , b , q ) -linear-free subsets of [ n ] , and define g k , b , q ( n ) = min S ∈ M k , b , q ( n ) | S | and f k , b , q ( n ) = max S ∈ M k , b , q ( n ) | S | . In this paper, formulae for g k , b , q ( n ) and f k , b , q ( n ) are proposed. Also, it is proven that there is at least one maximal ( k , b , q ) -linear-free subset of [ n ] with exactly x elements for any integer x between g k , b , q ( n ) and f k , b , q ( n ) , inclusively.

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4

Liu, H. Q. "On the distribution of k-free integers". Acta Mathematica Hungarica 144, nr 2 (18.10.2014): 269–84. http://dx.doi.org/10.1007/s10474-014-0454-9.

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5

Wlazinski, Francis. "A uniform cube-free morphism is k-power-free for all integers k ≥ 4". RAIRO - Theoretical Informatics and Applications 51, nr 4 (październik 2017): 205–16. http://dx.doi.org/10.1051/ita/2017015.

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6

Cellarosi, Francesco, i Ilya Vinogradov. "Ergodic properties of $k$-free integers in number fields". Journal of Modern Dynamics 7, nr 3 (2013): 461–88. http://dx.doi.org/10.3934/jmd.2013.7.461.

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7

Dong, D., i X. Meng. "Irrational Factor of Order k and ITS Connections With k-Free Integers". Acta Mathematica Hungarica 144, nr 2 (20.06.2014): 353–66. http://dx.doi.org/10.1007/s10474-014-0420-6.

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8

Choi, Dohoon, i Youngmin Lee. "Modular forms of half-integral weight on Γ₀(4) with few nonvanishing coefficients modulo ℓ". Open Mathematics 20, nr 1 (1.01.2022): 1320–36. http://dx.doi.org/10.1515/math-2022-0512.

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Abstract Let k k be a nonnegative integer. Let K K be a number field and O K {{\mathcal{O}}}_{K} be the ring of integers of K K . Let ℓ ≥ 5 \ell \ge 5 be a prime and v v be a prime ideal of O K {{\mathcal{O}}}_{K} over ℓ \ell . Let f f be a modular form of weight k + 1 2 k+\frac{1}{2} on Γ 0 {\Gamma }_{0} (4) such that its Fourier coefficients are in O K {{\mathcal{O}}}_{K} . In this article, we study sufficient conditions that if f f has the form f ( z ) ≡ ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v ) f\left(z)\equiv \mathop{\sum }\limits_{n=1}^{\infty }\mathop{\sum }\limits_{i=1}^{t}{a}_{f}\left({s}_{i}{n}^{2}){q}^{{s}_{i}{n}^{2}}\hspace{0.5em}\left({\rm{mod}}\hspace{0.33em}v) with square-free integers s i {s}_{i} , then f f is congruent to a linear combination of iterated derivatives of a single theta function modulo v v .

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9

LE BOUDEC, PIERRE. "POWER-FREE VALUES OF THE POLYNOMIAL t1⋯tr−1". Bulletin of the Australian Mathematical Society 85, nr 1 (23.09.2011): 154–63. http://dx.doi.org/10.1017/s0004972711002590.

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10

Benamar, Hela, Amara Chandoul i M. Mkaouar. "On the Continued Fraction Expansion of Fixed Period in Finite Fields". Canadian Mathematical Bulletin 58, nr 4 (1.12.2015): 704–12. http://dx.doi.org/10.4153/cmb-2015-055-9.

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AbstractThe Chowla conjecture states that if t is any given positive integer, there are infinitely many prime positive integers N such that Per() = t, where Per() is the period length of the continued fraction expansion for . C. Friesen proved that, for any k ∈ ℕ, there are infinitely many square-free integers N, where the continued fraction expansion of has a fixed period. In this paper, we describe all polynomials for which the continued fraction expansion of has a fixed period. We also give a lower bound of the number of monic, non-squares polynomials Q such that deg Q = 2d and Per =t.

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11

Aouissi, Siham, Moulay Chrif Ismaili, Mohamed Talbi i Abdelmalek Azizi. "Fields ℚ(d3,ζ3) whose 3-class group is of type (9,3)". International Journal of Number Theory 15, nr 07 (21.07.2019): 1437–47. http://dx.doi.org/10.1142/s1793042119500817.

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Let [Formula: see text] with [Formula: see text] a cube-free positive integer. Let [Formula: see text] be the 3-class group of k. With the aid of genus theory, arithmetic properties of the pure cubic field [Formula: see text] and some results on the 3-class group [Formula: see text], we determine all integers [Formula: see text] such that [Formula: see text].

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12

Martinek, Pavel. "On a Construction of Context-free Grammars". Fundamenta Informaticae 44, nr 3 (styczeń 2000): 245–64. https://doi.org/10.3233/fun-2000-44302.

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The grammatical inference problem is solved for the class of context-free languages. A context-free language is supposed to be given by means of all its strings. Considering all strings of length bounded by k, context-free grammars G_{j,k} with 1≤j<k are constructed. A~continual increasing of the index~$k$ leads to an~infinite sequence (G_{j,k})_{j<k}. It is proved that G_{j,k} are equivalent for all j≥j_0, k≥k_0 with some positive integers j_0, k_0. Moreover, the equivalences among these grammars can be recognized on the basis of involved nonterminals and productions only.

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13

Bilu, Yuri F., Jean-Marc Deshouillers, Sanoli Gun i Florian Luca. "Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms". Compositio Mathematica 154, nr 11 (18.10.2018): 2441–61. http://dx.doi.org/10.1112/s0010437x18007455.

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Let $\unicode[STIX]{x1D70F}(\cdot )$ be the classical Ramanujan $\unicode[STIX]{x1D70F}$-function and let $k$ be a positive integer such that $\unicode[STIX]{x1D70F}(n)\neq 0$ for $1\leqslant n\leqslant k/2$. (This is known to be true for $k<10^{23}$, and, conjecturally, for all $k$.) Further, let $\unicode[STIX]{x1D70E}$ be a permutation of the set $\{1,\ldots ,k\}$. We show that there exist infinitely many positive integers $m$ such that $|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(1))|<|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(2))|<\cdots <|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(k))|$. We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.

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14

Nyandwi, Servat. "Mean value of Piltz' function over integers free of large prime factors". Publications de l'Institut Math?matique (Belgrade) 74, nr 88 (2003): 37–56. http://dx.doi.org/10.2298/pim0374037n.

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We use the saddle-point method (due to Hildebrand?Tenenbaum [3]) to study the asymptotic behavior of ?n?x, p(n)?y ?k(n) for any k > 0 fixed, where P(n) is the greatest prime factor of n and ?k is Piltz' function. We generalize all results in [3], where the case k = 1 has been treated.

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15

Fagin, Barry. "Search Heuristics and Constructive Algorithms for Maximally Idempotent Integers". Information 12, nr 8 (29.07.2021): 305. http://dx.doi.org/10.3390/info12080305.

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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.

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16

Chakraborty, Kalyan, Shubham Gupta i Azizul Hoque. "Diophantine \(D(n)\)-quadruples in \(\mathbb{Z}[\sqrt{4k + 2}]\)". Glasnik Matematicki 59, nr 2 (15.12.2024): 259–76. https://doi.org/10.3336/gm.59.2.01.

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Let \(d\) be a square-free integer and \(\mathbb{Z}[\sqrt{d}]\) a quadratic ring of integers. For a given \(n\in\mathbb{Z}[\sqrt{d}]\), a set of \(m\) non-zero distinct elements in \(\mathbb{Z}[\sqrt{d}]\) is called a Diophantine \(D(n)\)-\(m\)-tuple (or simply \(D(n)\)-\(m\)-tuple) in \(\mathbb{Z}[\sqrt{d}]\) if product of any two of them plus \(n\) is a square in \(\mathbb{Z}[\sqrt{d}]\). Assume that \(d \equiv 2 \pmod 4\) is a positive integer such that \(x^2 - dy^2 = -1\) and \(x^2 - dy^2 = 6\) are solvable in integers. In this paper, we prove the existence of infinitely many \(D(n)\)-quadruples in \(\mathbb{Z}[\sqrt{d}]\) for \(n = 4m + 4k\sqrt{d}\) with \(m, k \in \mathbb{Z}\) satisfying \(m \not\equiv 5 \pmod{6}\) and \(k \not\equiv 3 \pmod{6}\). Moreover, we prove the same for \(n = (4m + 2) + 4k\sqrt{d}\) when either \(m \not\equiv 9 \pmod{12}\) and \(k \not\equiv 3 \pmod{6}\), or \(m \not\equiv 0 \pmod{12}\) and \(k \not\equiv 0 \pmod{6}\). At the end, some examples supporting the existence of quadruples in \(\mathbb{Z}[\sqrt{d}]\) with the property \(D(n)\) for the above exceptional \(n\)'s are provided for \(d = 10\).

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17

Hu, Yanan, i Zhenhua Lyu. "Two linear preserver problems on graphs". Electronic Journal of Linear Algebra 34 (21.02.2018): 602–8. http://dx.doi.org/10.13001/1081-3810.3792.

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Let n, t, k be integers such that 3 â¤ t,k â¤ n. Denote by G_n the set of graphs with vertex set {1,2,...,n}. In this paper, the complete linear transformations on G_n mapping K_t-free graphs to K_t-free graphs are characterized. The complete linear transformations on G_n mapping C_k-free graphs to C_k-free graphs are also characterized when n â¥ 6.

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18

Byott, N. P. "Some self-dual local rings of integers not free over their associated orders". Mathematical Proceedings of the Cambridge Philosophical Society 110, nr 1 (lipiec 1991): 5–10. http://dx.doi.org/10.1017/s0305004100070067.

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Let p be a prime number, and let K be a finite extension of the rational p-adic field ℚp. Let L/K be a finite abelian extension with Galois group G, and let L, K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG. Defining the associated order of the extension L/K byL can be viewed as a module over the ring , and a fortiori over the group ring KG.

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19

El Fadil, Lhoussain, i Omar Kchit. "On monogenity of certain pure number fields defined by $x^{2^r\cdot7^s}-m$". Boletim da Sociedade Paranaense de Matemática 41 (28.12.2022): 1–9. http://dx.doi.org/10.5269/bspm.62352.

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Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{2^r\cdot7^s}-m\in \mathbb{Z}[x]$, where $m\neq \pm 1$ is a square free integer, $r$ and $s$ are two positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md{4}$ and $\overline{m}\not\in\{\pm \overline{1},\pm \overline{18},\pm \overline{19}\} \md{49}$, then $K$ is monogenic. But if $r\geq 2$ and $m\equiv 1\md{16}$ or $s\geq 3$, $\overline{m}\in\{ \overline{1}, \overline{18}, -\overline{19}\} \md{49}$, and $\nu_7(m^6-1)\geq 4$, then $K$ is not monogenic. Some illustrating examples are given at the end of the paper.

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20

Plagne, Alain. "Maximal (k, l)-free sets in ℤ/pℤ are arithmetic progressions". Bulletin of the Australian Mathematical Society 65, nr 1 (luty 2002): 137–44. http://dx.doi.org/10.1017/s0004972700020153.

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Given two different positive integers k and l, a (k, l)-free set of some group (G, +) is defined as a set  ⊂ G such that k∩l = ∅. This paper is devoted to the complete determination of the structure of (k, l)-free sets of ℤ/pℤ (p an odd prime) with maximal cardinality. Except in the case where k = 2 and l = 1 (the so-called sum-free sets), these maximal sets are shown to be arithmetic progressions. This answers affirmatively a conjecture by Bier and Chin which appeared in a recent issue of this Bulletin.

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21

Shparlinski, Igor E. "On the Average Number of Square-Free Values of Polynomials". Canadian Mathematical Bulletin 56, nr 4 (1.12.2013): 844–49. http://dx.doi.org/10.4153/cmb-2012-021-8.

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Abstract.We obtain an asymptotic formula for the number of square-free integers in N consecutive values of polynomials on average over integral polynomials of degree at most k and of height at most H, where H ≥ Nk-1+ε for some fixed ε > 0. Individual results of this kind for polynomials of degree k > 3, due to A. Granville (1998), are only known under the ABC-conjecture.

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22

Shen, Yukai. "$ k $th powers in a generalization of Piatetski-Shapiro sequences". AIMS Mathematics 8, nr 9 (2023): 22411–18. http://dx.doi.org/10.3934/math.20231143.

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<abstract><p>The article considers a generalization of Piatetski-Shapiro sequences in the sense of Beatty sequences. The sequence is defined by $ \left(\left\lfloor\alpha n^c+\beta\right\rfloor\right)_{n = 1}^{\infty} $, where $ \alpha \geq 1 $, $ c > 1 $, and $ \beta $ are real numbers.</p> <p>The focus of the study is on solving equations of the form $ \left\lfloor \alpha n^c +\beta\right\rfloor = s m^k $, where $ m $ and $ n $ are positive integers, $ 1 \leq n \leq N $, and $ s $ is an integer. Bounds for the solutions are obtained for different values of the exponent $ k $, and an average bound is derived over $ k $-free numbers $ s $ in a given interval.</p></abstract>

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23

Patterson, Cody, Kirby C. Smith i Leon Van Wyk. "Factor rings of the Gaussian integers". Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 23, nr 4 (23.09.2004): 114–25. http://dx.doi.org/10.4102/satnt.v23i4.201.

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Whereas the homomorphic images of Z (the ring of integers) are well known, namely Z, {0} and Zn (the ring of integers modulo n), the same is not true for the homomorphic im-ages of Z[i] (the ring of Gaussian integers). More generally, let m be any nonzero square free integer (positive or negative), and consider the integral domain Z[ √m]={a + b √m | a, b ∈ Z}. Which rings can be homomorphic images of Z[ √m]? This ques-tion oﬀers students an inﬁnite number (one for each m) of investigations that require only undergraduate mathematics. It is the goal of this article to oﬀer a guide to the in-vestigation of the possible homomorphic images of Z[ √m] using the Gaussian integers Z[i] as an example. We use the fact that Z[i] is a principal ideal domain to prove that if I =(a+bi) is a nonzero ideal of Z[i], then Z[i]/I ∼ = Zn for a positive integer n if and only if gcd{a, b} =1, in which case n = a2 + b2 . Our approach is novel in that it uses matrix techniques based on the row reduction of matrices with integer entries. By characterizing the integers n of the form n = a2 + b2 , with gcd{a, b} =1, we obtain the main result of the paper, which asserts that if n ≥ 2, then Zn is a homomorphic image of Z[i] if and only if the prime decomposition of n is 2α0 pα1 1 ··· pαk k , with α0 ∈{0, 1},pi ≡ 1(mod 4) and αi ≥ 0 for every i ≥ 1. All the ﬁelds which are homomorpic images of Z[i] are also determined.

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24

Broughan, Kevin A. "Adic Topologies for the Rational Integers". Canadian Journal of Mathematics 55, nr 4 (1.08.2003): 711–23. http://dx.doi.org/10.4153/cjm-2003-030-3.

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AbstractA topology on ℤ, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to ℚ, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on ℤ, which includes the p-adics, and one in which the set of rational primes ℙ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and k-free numbers.

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25

Robbins, Neville. "On the Number of Binomial Coefficients Which are Divisible by their Row Number: II". Canadian Mathematical Bulletin 28, nr 4 (1.12.1985): 481–86. http://dx.doi.org/10.4153/cmb-1985-059-0.

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AbstractIf n is a natural number, let A(n) denote the number of integers, k, such that 0 < k < n and n divides . Let ϕ(n) denote Euler's totient function. Necessary and sufficient conditions are given so that A(n) = ϕ(n) when n is square-free.

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26

Snaith, Victor. "Restricted Determinantal Homomorphisms and Locally Free Class Groups". Canadian Journal of Mathematics 42, nr 4 (1.08.1990): 646–58. http://dx.doi.org/10.4153/cjm-1990-034-7.

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Let K be a number field and let OK denote the integers of K. The locally free class groups, Cl(OK[G]), furnish a fundamental collection of invariants of a finite group, G. In this paper I will construct some new, non-trivial homomorphisms, called restricted determinants, which map the NGH-invariant idèlic units of Ok([Hab] to Cl(OK[G]). These homomorphisms are constructed by means of the Horn-description of Cl(OK[G]), which describes the locally free class group in terms of the representation theory of G, and the technique of Explicit Brauer Induction, which was introduced in [5].

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27

Bard, Stefan, Gary MacGillivray i Shayla Redlin. "The complexity of frugal colouring". Arabian Journal of Mathematics 10, nr 1 (29.01.2021): 51–57. http://dx.doi.org/10.1007/s40065-021-00311-7.

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AbstractA t-frugal colouring of a graph G is an assignment of colours to the vertices of G, such that each colour appears at most t times in the neighbourhood of any vertex. A dichotomy theorem for the complexity of deciding whether a graph has a 1-frugal colouring with k colours was found by McCormick and Thomas, and then later extended to restricted graph classes by Kratochvil and Siggers. We generalize the McCormick and Thomas theorem by proving a dichotomy theorem for the complexity of deciding whether a graph has a t-frugal colouring with k colours, for all pairs of positive integers t and k. We also generalize bounds of Lih et al. for the number of colours needed in a 1-frugal colouring of a given $$K_4$$ K 4 -minor-free graph with maximum degree $$\Delta $$ Δ to t-frugal colourings, for any positive integer t.

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28

Gevorkyan, Yuriy. "Geometric approach to the proof of Fermat’s last theorem". EUREKA: Physics and Engineering, nr 4 (30.07.2022): 127–36. http://dx.doi.org/10.21303/2461-4262.2022.002488.

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A geometric approach to the proof of Fermat’s last theorem is proposed. Instead of integers a, b, c, Fermat’s last theorem considers a triangle with side lengths a, b, c. It is proved that in the case of right-angled and obtuse-angled triangles Fermat's equation has no solutions. When considering the case when a, b, c are sides of an acute triangle, it is proved that Fermat's equation has no entire solutions for p>2. The numbers a=k, b=k+m, c=k+n, where k, m, n are natural numbers satisfying the inequalities n>m, n<k+m, exhaust all possible variants of natural numbers a, b, c which are the sides of the triangle. The proof in this case is carried out by introducing a new auxiliary function f(k,p)=kp+(k+m)p–(k+n)p of two variables, which is a polynomial of degree in the variable . The study of this function necessary for the proof of the theorem is carried out. A special case of Fermat’s last theorem is proved, when the variables a, b, c take consecutive integer values. The proof of Fermat’s last theorem was carried out in two stages. Namely, all possible values of natural numbers k, m, n, p were considered, satisfying the following conditions: firstly, the number (np–mp) is odd, and secondly, this number is even, where the number (np–mp) is a free member of the function f(k, p). Another proof of Fermat’s last theorem is proposed, in which all possible relationships between the supposed integer solution of the equation f(k, p)=0 and the number corresponding to this supposed solution are considered. The proof is carried out using the mathematical apparatus of the theory of integers, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author's works, in which some special cases of Fermat’s last theorem are proved

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29

Li, Xin. "K-theory for ring C*-algebras attached to function fields with only one infinite place". Journal of K-Theory 10, nr 1 (31.01.2012): 203–31. http://dx.doi.org/10.1017/is011011023jkt177.

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AbstractWe study the K-theory of ring C*-algebras associated to rings of integers in global function fields with only a single infinite place. First, we compute the torsion-free part of the K-groups of these ring C*-algebras. Secondly, we show that, under a certain primeness condition, the torsion part of K-theory determines the inertia degrees at infinity of our function fields.

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30

Grósz, Dániel, Abhishek Methuku i Casey Tompkins. "On subgraphs of C2k-free graphs and a problem of Kühn and Osthus". Combinatorics, Probability and Computing 29, nr 3 (4.02.2020): 436–54. http://dx.doi.org/10.1017/s0963548319000452.

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AbstractLet c denote the largest constant such that every C6-free graph G contains a bipartite and C4-free subgraph having a fraction c of edges of G. Győri, Kensell and Tompkins showed that 3/8 ⩽ c ⩽ 2/5. We prove that c = 38. More generally, we show that for any ε > 0, and any integer k ⩾ 2, there is a C2k-free graph $G'$ which does not contain a bipartite subgraph of girth greater than 2k with more than a fraction $$\Bigl(1-\frac{1}{2^{2k-2}}\Bigr)\frac{2}{2k-1}(1+\varepsilon)$$ of the edges of $G'$ . There also exists a C2k-free graph $G''$ which does not contain a bipartite and C4-free subgraph with more than a fraction $$\Bigl(1-\frac{1}{2^{k-1}}\Bigr)\frac{1}{k-1}(1+\varepsilon)$$ of the edges of $G''$ .One of our proofs uses the following statement, which we prove using probabilistic ideas, generalizing a theorem of Erdős. For any ε > 0, and any integers a, b, k ⩾ 2, there exists an a-uniform hypergraph H of girth greater than k which does not contain any b-colourable subhypergraph with more than a fraction $$\Bigl(1-\frac{1}{b^{a-1}}\Bigr)(1+\varepsilon)$$ of the hyperedges of H. We also prove further generalizations of this theorem.In addition, we give a new and very short proof of a result of Kühn and Osthus, which states that every bipartite C2k-free graph G contains a C4-free subgraph with at least a fraction 1/(k−1) of the edges of G. We also answer a question of Kühn and Osthus about C2k-free graphs obtained by pasting together C2l’s (with k >l ⩾ 3).

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31

Jakimczuk, Rafael. "Mertens's formula, k-full numbers and sum of two squares". Gulf Journal of Mathematics 17, nr 2 (27.09.2024): 263–91. http://dx.doi.org/10.56947/gjom.v17i2.2290.

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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.

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32

Tang, Betty, Solomon W. Golomb i Ronald L. Graham. "A New Result on Comma-Free Codes of Even Word-Length". Canadian Journal of Mathematics 39, nr 3 (1.06.1987): 513–26. http://dx.doi.org/10.4153/cjm-1987-023-7.

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Comma-free codes were first introduced in [1] in 1957 as a possible genetic coding scheme for protein synthesis. The general mathematical setting of such codes was presented in [3], and the biochemical and mathematical aspects of the problem were later summarized and extended in [4].Using the notation of [3], a set D of k-tuples or k-letter words, (a1a2 … ak), wherefor fixed positive integers k and n, is said to be a comma-free dictionary if and only if, whenever (a1a2 … ak) and (b1b2 … bk) are in D, the “overlaps”are not in D. This precludes codewords having a subperiod less than k; and two codewords which are cyclic permutations of one another cannot both be in D.

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33

Czédli, Gábor. "Minimum-sized generating sets of the direct powers of free distributive lattices". Cubo (Temuco) 26, nr 2 (28.06.2024): 217–37. http://dx.doi.org/10.56754/0719-0646.2602.217.

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For a finite lattice \(L\), let Gm(\(L\)) denote the least \(n\) such that \(L\) can be generated by \(n\) elements. For integers \(r>2\) and \(k>1\), denote by FD\((r)^k\) the \(k\)-th direct power of the free distributive lattice FD(\(r\)) on \(r\) generators. We determine Gm(FD\((r)^k\)) for many pairs \((r,k)\) either exactly or with good accuracy by giving a lower estimate that becomes an upper estimate if we increase it by 1. For example, for \((r,k)=(5,25\,000)\) and \((r,k)=(20,\ 1.489\cdot 10^{1789})\), Gm(FD\((r)^k\)) is \(22\) and \(6\,000\), respectively. To reach our goal, we give estimates for the maximum number of pairwise unrelated copies of some specific posets (called full segment posets) in the subset lattice of an \(n\)-element set. In addition to analogous earlier results in lattice theory, a connection with cryptology is also mentioned among the motivations.

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34

Halberstam, H. "Application of a method of Szemeredi". Glasgow Mathematical Journal 27 (październik 1985): 81–85. http://dx.doi.org/10.1017/s001708950000608x.

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Let ℬ = {bi:b1 <b2<…} be an infinite sequence of positive integers that exceed 1 and are pairwise coprime, so thatAssume also thatLet A=Aℬ denote the sequence of ℬ-free numbers, that is, of positive integers divisible by no element of ℬ. This concept, generalizing square-free and k-free numbers, derives from Erdös [2] who proved in 1966 that there exists a constant c, 0<c<l, independent of ℬ, such that the interval (x, x+xc) contains elements of A provided only that x is large enough. This result of Erdös was shown by Szemeredi [7] in 1973 to hold with c=½+ε, if x≥xo(ε, ℬ), and quite recently Bantle and Grupp [1] have sharpened Szemeredi's result to c=9/20+ε.

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35

Bley, Werner, i Stephen M. J. Wilson. "Computations in Relative Algebraic K-Groups". LMS Journal of Computation and Mathematics 12 (2009): 166–94. http://dx.doi.org/10.1112/s1461157000001480.

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Let G be finite group and K a number field or a p-adic field with ring of integers OK. In the first part of the manuscript we present an algorithm that computes the relative algebraic K-group K0(OK[G], K) as an abstract abelian group. We also give algorithms to solve the discrete logarithm problems in K0(OK[G], K) and in the locally free class group cl(OK[G]). All algorithms have been implemented in Magma for the case K = Q.In the second part of the manuscript we prove formulae for the torsion subgroup of K0(Z[G], Q) for large classes of dihedral and quaternion groups.

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36

Bodini, Olivier. "On the strange kinetic aesthetic of rectangular shape partitions". Pure Mathematics and Applications 30, nr 1 (1.06.2022): 37–44. http://dx.doi.org/10.2478/puma-2022-0007.

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Abstract In this paper, we focus on shape partitions. We show that for any fixed k, one can symbolically characterize the shape partition on a k × n rectangular grid by a context-free grammar. We explicitly give this grammar for k = 2 and k = 3 (for k = 1, this corresponds to compositions of integers). From these grammars, we deduce the number of shape partitions for the k × n rectangular grids for k ∈ {1, 2, 3} and every n, as well as the limiting Gaussian distribution of the number of connected components. This also enables us to randomly and uniformly generate shape partitions of large size.

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37

E. Charkani, Mohammed, i Omar Boughaleb. "On the ideal discriminant of some relative pure extensions". Boletim da Sociedade Paranaense de Matemática 42 (22.05.2024): 1–7. http://dx.doi.org/10.5269/bspm.64902.

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Let L = K(α) be an extension of a number field K where α satisfies the monic irreducible polynomial P(X) = Xp −a ∈ R[X] of prime degree p and such that a is pth power free in R := OK (the ring of integers of K). The purpose of this paper is to give an explicit formula for the ideal discriminant DL/K of L over K involving only the prime ideals dividing the principal ideals aR and pR. As an illustration, we compute the discriminant DL/K of a family of septic and quintic pure fields over quadratic fields. Hence a slightly simpler computation of discriminant DL/K is obtained.

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38

KAUR, INDER. "A pathological case of the C1 conjecture in mixed characteristic". Mathematical Proceedings of the Cambridge Philosophical Society 167, nr 01 (5.04.2018): 61–64. http://dx.doi.org/10.1017/s0305004118000178.

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AbstractLet K be a field of characteristic 0. Fix integers r, d coprime with r ⩾ 2. Let XK be a smooth, projective, geometrically connected curve of genus g ⩾ 2 defined over K. Assume there exists a line bundle ${\cal L}_K$ on XK of degree d. In this paper we prove the existence of a stable locally free sheaf on XK with rank r and determinant ${\cal L}_K$. This trivially proves the C1 conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.

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39

PAPISTAS, A. I. "AUTOMORPHISMS OF CERTAIN RELATIVELY FREE GROUPS AND LIE ALGEBRAS". International Journal of Algebra and Computation 14, nr 03 (czerwiec 2004): 311–23. http://dx.doi.org/10.1142/s0218196704001761.

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For positive integers n and c, with n≥2, let Gn,c be a relatively free group of rank n in the variety N2A∧AN2∧Nc. It is shown that there exists an explicitly described finite subset Ω of IA-automorphisms of Gn,c such that the cardinality of Ω is independent upon n and c and the subgroup of the automorphism group Aut (Gn,c) of Gn,c generated by the tame automorphisms and Ω has finite index in Aut (Gn,c). This is a simpler result than one given in [12, Theorem 1(I)]. Let L(Gn,c) be the associated Lie ring of Gn,c and K be a field of characteristic zero. The method developed in the proof of the aforementioned result is applied in order to find an explicitly described finite subset ΩL of the IA-automorphism group of K⊗L(Gn,c) such that the automorphism group of K⊗L(Gn,c) is generated by GL (n,K) and ΩL. In particular, for n≥3, the cardinality of ΩL is independent upon n and c.

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40

KIM, SEONG KUN. "ON THE ASPHERICITY OF CERTAIN RELATIVE PRESENTATIONS OVER TORSION-FREE GROUPS". International Journal of Algebra and Computation 18, nr 06 (wrzesień 2008): 979–87. http://dx.doi.org/10.1142/s0218196708004718.

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It is noteworthy to find whether a relative presentation with torsion-free coefficients is aspherical or not, since a nonaspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture, see [7]. In this paper we investigate asphericity of the relative group presentation [Formula: see text] provided that m1, m2, …, mk are nonzero integers, k ≤ 3, each coefficient has infinite order and the relator is not a proper power.

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41

KACZOROWSKI, J., i A. PERELLI. "On the distribution in short intervals of products of a prime and integers from a given set". Mathematical Proceedings of the Cambridge Philosophical Society 124, nr 1 (lipiec 1998): 1–14. http://dx.doi.org/10.1017/s0305004197002284.

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A classical problem in analytic number theory is the distribution in short intervals of integers with a prescribed multiplicative structure, such as primes, almost-primes, k-free numbers and others. Recently, partly due to applications to cryptology, much attention has been received by the problem of the distribution in short intervals of integers without large prime factors, see Lenstra–Pila–Pomerance [3] and section 5 of the excellent survey by Hildebrand–Tenenbaum [1].In this paper we deal with the distribution in short intervals of numbers representable as a product of a prime and integers from a given set [Sscr ], defined in terms of cardinality properties. Our results can be regarded as an extension of the above quoted results, and we will provide a comparison with such results by a specialization of the set [Sscr ].

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42

Xuan, Ti. "The average order of $d_{k}(n)$ over integers free of large prime factors". Acta Arithmetica 55, nr 3 (1990): 249–60. http://dx.doi.org/10.4064/aa-55-3-249-260.

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43

Miyata, Yoshimasa. "Vertices of ideals of a -adic number field II". Nagoya Mathematical Journal 107 (wrzesień 1987): 49–62. http://dx.doi.org/10.1017/s0027763000002531.

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Let k be a -adic number field with the ring 0 of all integers in k, and K be a finite normal extension with Galois group G. ∏ denotes a prime element of the ring of all integers in K. Then, an ideal (∏a) of is an 0G-module. E. Noether showed that if K/k is tamely ramified, is a free 0G-module. A. Fröhlich generalized E. Noether’s theorem as follows: is relatively projective with respect to a subgroup S of G if and only if S ⊇ G1 where G1 is the first ramification group of K/k. Now we define the vertex V(∏a) of (∏a) as the minimal normal subgroup S of G such that (∏a) is relatively projective with respect to a subgroup S of G (cf. § 1). Then, the above generalization by A. Fröhlich implies V() = G1. In the previous paper, we proved G1 ⊇ V(∏a) ⊇ G2, (where G2 is the second ramification group of K/k (cf. Theorem 5). Further, we dealt with the case where G = G1 is of order p2, and proved that if V(∏a) ≠ G1 then a ≡ 1(p2) and t2 ≡ 1(p2) for the second ramification number t2 of K/k (cf. Theorems 15 and 21). The purpose of this paper is to prove the similar theorem for the wildly ramified p-extension of degree pn (Theorem 7).

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44

Saikia, A. "Selmer Groups of Elliptic Curves with Complex Multiplication". Canadian Journal of Mathematics 56, nr 1 (1.02.2004): 194–208. http://dx.doi.org/10.4153/cjm-2004-009-7.

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AbstractSuppose K is an imaginary quadratic field and E is an elliptic curve over a number field F with complex multiplication by the ring of integers in K. Let p be a rational prime that splits as in K. Let Epn denote the pn-division points on E. Assume that F(Epn) is abelian over K for all n ≥ 0. This paper proves that the Pontrjagin dual of the -Selmer group of E over F(Ep∞) is a finitely generated free Λ-module, where Λ is the Iwasawa algebra of . It also gives a simple formula for the rank of the Pontrjagin dual as a Λ-module.

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45

Laubenbacher, Reinhard C., i Bruce A. Magurn. "Sk2 and K3 Of Dihedral Groups". Canadian Journal of Mathematics 44, nr 3 (1.06.1992): 591–623. http://dx.doi.org/10.4153/cjm-1992-037-x.

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AbstractNew computations of birelative K2 groups and recent results on K3 of rings of algebraic integers are combined in generalized Mayer-Vietoris sequences for algebraic k-theory. Upper and lower bounds for SK2(ℤ G) and lower bounds for K3(ℤ G) are deduced for G a dihedral group of square-free order, and for some other closely related groups G.

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46

Nicklasson, Lisa. "On the Betti numbers and Rees algebras of ideals with linear powers". Journal of Algebraic Combinatorics 53, nr 2 (marzec 2021): 575–92. http://dx.doi.org/10.1007/s10801-021-01026-w.

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AbstractAn ideal $$I \subset \mathbb {k}[x_1, \ldots , x_n]$$ I ⊂ k [ x 1 , … , x n ] is said to have linear powers if $$I^k$$ I k has a linear minimal free resolution, for all integers $$k>0$$ k > 0 . In this paper, we study the Betti numbers of $$I^k$$ I k , for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for $$d=2, 3$$ d = 2 , 3 or $$n-1$$ n - 1 , and the product of all ideals generated by s variables, for $$s=n-1$$ s = n - 1 or $$n-2$$ n - 2 . We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.

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47

Liu, Yanjiao, i Jianhua Yin. "The generalized Turán number of $ 2 S_\ell $". AIMS Mathematics 8, nr 10 (2023): 23707–12. http://dx.doi.org/10.3934/math.20231205.

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<abstract><p>The generalized Turán number $ ex{(n, K_s, H)} $ is defined to be the maximum number of copies of a complete graph $ K_s $ in any $ H $-free graph on $ n $ vertices. Let $ S_\ell $ denote the star on $ \ell+1 $ vertices, and let $ kS_\ell $ denote the disjoint union of $ k $ copies of $ S_\ell $. Gan et al. and Chase determined $ ex(n, K_s, S_\ell) $ for all integers $ s\ge 3 $, $ \ell\ge 1 $ and $ n\ge 1 $. In this paper, we determine $ ex(n, K_s, 2S_\ell) $ for all integers $ s\ge 4 $, $ \ell\ge 1 $ and $ n\ge 1 $.</p></abstract>

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48

KUO, WENTANG, i DAVID TWEEDLE. "Primitive submodules for Drinfeld modules". Mathematical Proceedings of the Cambridge Philosophical Society 159, nr 2 (30.06.2015): 275–302. http://dx.doi.org/10.1017/s0305004115000353.

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AbstractThe ring A = $\mathbb{F}$r[T] and its fraction field k, where r is a power of a prime p, are considered as analogues of the integers and rational numbers respectively. Let K/k be a finite extension and let φ be a Drinfeld A-module over K of rank d and Γ ⊂ K be a finitely generated free A-submodule of K, the A-module structure coming from the action of φ. We consider the problem of determining the number of primes ℘ of K for which the reduction of Γ modulo ℘ is equal to $\mathbb{F}$℘ (the residue field of the prime ℘). We can show that there is a natural density of primes ℘ for which Γ mod ℘ is equal to $\mathbb{F}$℘. In certain cases, this density can be seen to be positive.

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49

AGBOOLA, A. "On p-adic height pairings and locally free classgroups of Hopf orders". Mathematical Proceedings of the Cambridge Philosophical Society 123, nr 3 (maj 1998): 447–59. http://dx.doi.org/10.1017/s0305004197002296.

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Let E be an elliptic curve with complex multiplication by the ring of integers [Ofr ] of an imaginary quadratic field K. The purpose of this paper is to describe certain connections between the arithmetic of E on the one hand and the Galois module structure of certain arithmetic principal homogeneous spaces arising from E on the other. The present paper should be regarded as a complement to [AT]; we assume that the reader is equipped with a copy of the latter paper and that he is not averse to referring to it from time to time.

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50

Waters, L. K. "THE UNIQUE EQUIVALENCE CLASS OF CHROMATIC RECTANGLE FREE 3-COLORINGS OF A 10 × 10 CHESSBOARD". Asian-European Journal of Mathematics 03, nr 04 (grudzień 2010): 715–29. http://dx.doi.org/10.1142/s1793557110000544.

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Consider an M × N chess board with each space colored one of K colors. A chromatic rectangle is a rectangular collection of spaces with all four corner spaces colored the same. An M × N : K NCR board is an M × N board for which there exists a K coloring with no chromatic rectangles. If every K coloring includes a chromatic rectangle, then that board is called an M × N : K CR board. The classification as NCR versus CR has been settled for K ∈ {1, 2, 3} and all positive integers N and M. Note that transposition, or interchanging rows, columns, or colors, will preserve the existence of chromatic rectangles within a coloring. With this in mind, two colorings of a board are called equivalent if one can be produced from the other by such manipulations. This paper establishes that all 10 × 10 : 3 NCR colorings are equivalent. The results stem from characterizations of NCR colorings. These characterizations permit devising and implementing a backtracking algorithm for finding NCR colorings within a significantly restricted search space. In the 10 × 10 : 3 case, the restricted search space is small enough to complete an exhaustive search in about an hour. Several NCR colorings for larger boards, with K > 3, are also included.

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