共查询到20条相似文献,搜索用时 15 毫秒
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We consider arithmetic progressions consisting of integers which are y-components of solutions of an equation of the form x 2 ? dy 2 = m. We show that for almost all four-term arithmetic progressions such an equation exists. We construct a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different equations of the given form. These results are obtained by studying the properties of a parametric family of elliptic curves. 相似文献
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We give a complete characterization of so-called powerful arithmetic progressions, i.e. of progressions whose kth term is a kth power for all k. We also prove that the length of any primitive arithmetic progression of powers can be bounded both by any term of the progression different from 0 and ±1, and by its common difference. In particular, such a progression can have only finite length. 相似文献
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We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved by Aharoni and Berger in 2009.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays. 相似文献
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A positive integer n is called a square-full number if p 2 divides n whenever p is a prime divisor of n. In this paper we study the distribution of square-full numbers in arithmetic progressions by using the properties of Riemann zeta functions and Dirichlet L-functions. 相似文献
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Y. Zhang 《Acta Mathematica Hungarica》2018,156(1):240-254
Let \({f(x, k, d) = x(x + d)\cdots(x + (k - 1)d)}\) be a polynomial with \({k \geq 2}\), \({d \geq 1}\). We consider the Diophantine equation \({\prod_{i = 1}^{r} f(x_i, k_i, d) = y^2}\), which is inspired by a question of Erd?s and Graham [4, p. 67]. Using the theory of Pellian equation, we give infinitely many (nontrivial) positive integer solutions of the above Diophantine equation for some cases. 相似文献
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The 2-color Rado number for the equation x1+x2−2x3=c, which for each constant
we denote by S1(c), is the least integer, if it exists, such that every 2-coloring, Δ : [1,S1(c)]→{0,1}, of the natural numbers admits a monochromatic solution to x1+x2−2x3=c, and otherwise S1(c)=∞. We determine the 2-color Rado number for the equation x1+x2−2x3=c, when additional inequality restraints on the variables are added. In particular, the case where we require x2<x3<x1, is a generalization of the 3-term arithmetic progression; and the work done here improves previously established upper bounds to an exact value. 相似文献
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Yoichi Motohashi 《Inventiones Mathematicae》1978,44(2):163-178
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Igor E. Shparlinski 《Periodica Mathematica Hungarica》2013,67(1):55-61
We show that for any integers a and m with m ≥ 1 and gcd(a,m) = 1, there is a solution to the congruence pr ≡ a (modm) where p is prime, r is a product of at most k = 17 prime factors and p, r ≤ m. This is a relaxed version of the still open question, studied by P. Erd?s, A. M. Odlyzko and A. Sárközy, that corresponds to k = 1 (that is, to products of two primes). 相似文献
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The Ramanujan Journal - Recently, Mc Laughlin proved some results on vanishing coefficients in the series expansions of certain infinite q-products for arithmetic progressions modulo 5, modulo 7... 相似文献
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Recent finiteness results concerning the lengths of arithmetic progressions in linear combinations of elements from finitely
generated multiplicative groups have found applications to a variety of problems in number theory. In the present paper, we
significantly refine the existing arguments and give an explicit upper bound on the length of such progressions. 相似文献
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Estimating the discrepancy of the set of all arithmetic progressions in the first N natural numbers was one of the famous open problems in combinatorial discrepancy theory for a long time, successfully solved by K. Roth (lower bound) and Beck (upper bound). They proved that D(N)=minχmaxA|∑xAχ(x)|=Θ(N1/4), where the minimum is taken over all colorings χ:[N]→{−1,1} and the maximum over all arithmetic progressions in [N]={0,…,N−1}.Sumsets of k arithmetic progressions, A1++Ak, are called k-arithmetic progressions and they are important objects in additive combinatorics. We define Dk(N) as the discrepancy of the set {P∩[N]:P is a k-arithmetic progression}. The second author proved that Dk(N)=Ω(Nk/(2k+2)) and Přívětivý improved it to Ω(N1/2) for all k≥3. Since the probabilistic argument gives Dk(N)=O((NlogN)1/2) for all fixed k, the case k=2 remained the only case with a large gap between the known upper and lower bounds. We bridge this gap (up to a logarithmic factor) by proving that Dk(N)=Ω(N1/2) for all k≥2.Indeed we prove the multicolor version of this result. 相似文献
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Pablo A. Parrilo 《Journal of Combinatorial Theory, Series A》2008,115(1):185-192
Let V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-coloring of {1,2,…,n}. We show that
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Letm 3 andk 1 be two given integers. Asub-k-coloring of [n] = {1, 2,...,n} is an assignment of colors to the numbers of [n] in which each color is used at mostk times. Call an
arainbow set if no two of its elements have the same color. Thesub-k-Ramsey number sr(m, k) is defined as the minimumn such that every sub-k-coloring of [n] contains a rainbow arithmetic progression ofm terms. We prove that((k – 1)m
2/logmk) sr(m, k) O((k – 1)m
2 logmk) asm , and apply the same method to improve a previously known upper bound for a problem concerning mappings from [n] to [n] without fixed points.Research supported in part by Allon Fellowship and by a Bat Sheva de-Rothschild grant.Research supported in part by the AKA Research Fund of the Hungarian Academy of Sciences, grant No. 1-3-86-264. 相似文献
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Emre Alkan Andrew H. Ledoan Marian Vâjâitu Alexandru Zaharescu 《The Ramanujan Journal》2008,16(2):131-161
We prove asymptotic formulas for the first and second moments of the index of fractions with square-free denominators of order Q streaming in a given arithmetic progression as Q→∞. A. Zaharescu was supported by NSF grant number DMS-0456615. This research was also partially supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research. 相似文献
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József Beck 《Journal of Combinatorial Theory, Series A》1980,29(3):376-379
F. Cohen raised the following question: Determine or estimate a function F(d) so that if we split the integers into two classes at least one class contains, for infinitely many values of d, an arithmetic progression of difference d and length F(d). We prove F(d) ? (1 + ε) log2d. 相似文献
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Frank W Schmidt 《Journal of Combinatorial Theory, Series A》1982,33(1):30-35
The density of sets not containing a diagonal, a special type of arithmetic progression, is investigated. A lower bound on this density is established which sharpens a result of Alspach, Brown, and Hell (J. London Math. Soc.13 (2) (1976), 226–334. 相似文献