Showing papers on "Arithmetic progression published in 2011"
TL;DR: A positive definite quadratic form f is said to be regular if it globally represents all integers that are represented by the genus of f. In as discussed by the authors, the authors provided a list of 913 candidates of primitive positive definite regular ternary Quadratic forms.
Abstract: A positive definite quadratic form f is said to be regular if it globally represents all integers that are represented by the genus of f . In 1997, Jagy, Kaplansky and Schiemann provided a list of 913 candidates of primitive positive definite regular ternary quadratic forms, and all but 22 of them are verified to be regular. In this talk we show that 8 forms among 22 candidates are, in fact, regular. At the end of the talk, we show some finiteness result on ternary forms that represent every eligible integer in some arithmetic progression.
60 citations
TL;DR: For numerical semigroups, this article gave an affirmative answer to Wilf's conjecture for numerical semiigroups satisfying 2
u \geq m, where
u and m are respectively the embedding dimension and the multiplicity of a semigroup.
Abstract: We give an affirmative answer to Wilf's conjecture for numerical semigroups satisfying 2
u \geq m, where
u and m are respectively the embedding dimension and the multiplicity of a semigroup. The conjecture is also proved when m \leq 8 and when the semigroup is generated by a generalized arithmetic sequence.
39 citations
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TL;DR: In this article, the L^p-almost-periodicity of convolutions was shown to hold for any subsets of {1,...,N} of sizes alpha N and beta N, and the sumset A+B of the subsets A and B has an arithmetic progression of length at least exp(c (c (alpha beta log N)^{1/2}).
Abstract: We prove results about the L^p-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L^p, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,...,N} of sizes alpha N and beta N then A+B contains an arithmetic progression of length at least about exp(c (alpha beta log N)^{1/2}). Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least about exp(c (alpha log N/(log(beta^{-1}))^3)^{1/2}).
31 citations
TL;DR: In this article, a generalization of the Frobenius problem is studied, where given a positive relatively prime integer, what is the largest integer that cannot be represented as a nonnegative integral linear combination of these parameters?
Abstract: We study a generalization of the \emph{Frobenius problem}: given $k$ positive relatively prime integers, what is the largest integer $g_0$ that cannot be represented as a nonnegative integral linear combination of these parameters? More generally, what is the largest integer $g_s$ that has exactly $s$ such representations? We illustrate a family of parameters, based on a recent paper by Tripathi, whose generalized Frobenius numbers $g_0, \ g_1, \ g_2, ...$ exhibit unnatural jumps; namely, $g_0, \ g_1, \ g_k, \ g_{\binom{k+1}{k-1}}, \ g_{\binom{k+2}{k-1}}, ...$ form an arithmetic progression, and any integer larger than $g_{\binom{k+j}{k-1}}$ has at least $\binom{k+j+1}{k-}$ representations. Along the way, we introduce a variation of a generalized Frobenius number and prove some basic results about it.
30 citations
TL;DR: In this paper, the problem of labeling the vertices, edges and faces of a plane graph was studied and a weight of a face is defined as the sum of the label of the face and the labels of vertices and edges surrounding that face.
Abstract: This paper deals with the problem of labeling the vertices, edges and faces of a plane graph. A weight of a face is the sum of the label of a face and the labels of the vertices and edges surrounding that face. In a super d-antimagic labeling the vertices receive the smallest labels and the weights of all s-sided faces constitute an arithmetic progression of difference d, for each s appearing in the graph.
24 citations
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TL;DR: In this paper, the sharp asymptotic behavior of c(p) as p + √ √ n + n is established for all arithmetic progressions of nonnegative integers, and generalizations and variations of this are also established.
Abstract: Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set {T {\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues (respectively, non-residues) of p}. When S is constructed in various ways from the set of all arithmetic progressions of nonnegative integers, we determine the sharp asymptotic behavior of c(p) as p {\to} +{\infty}. Generalizations and variations of this are also established, and some problems connected with these results that are worthy of further study are discussed.
18 citations
TL;DR: For a given set M of positive integers, a problem of Motzkin this paper asks for determining the maximal density μ (M ) among sets of nonnegative integers in which no two elements differ by an element of M. The problem is completely settled when |M | ⩽ 2, and some partial results are known for several families of M for |M| ⩾ 3, including the case where the elements of M are in arithmetic progression.
17 citations
8 Feb 2011
TL;DR: An infinite number of Edwards curves with an arithmetic progression of length 9 is found, which means that the x-coordinates of a sequence of rational points on the curve form an arithmetic progressions.
Abstract: We look at arithmetic progressions on elliptic curves known as Edwards curves. By an arithmetic progression on an elliptic curve, we mean that the x-coordinates of a sequence of rational points on the curve form an arithmetic progression. Previous work has found arithmetic progressions on Weierstrass curves, quartic curves, and genus 2 curves. We find an infinite number of Edwards curves with an arithmetic progression of length 9.
16 citations
TL;DR: In this article, the signed partitions of zero in arithmetic progression are examined with intent to expand the standard repertoire of mappings and statistics used in bijective proofs for ordinary partition identities.
Abstract: The recently formalized idea of signed partitions is examined with intent to expand the standard repertoire of mappings and statistics used in bijective proofs for ordinary partition identities. A new family of partitions is added to Schur’s Theorem and observations are made concerning the behavior of signed partitions of zero in arithmetic progression.
14 citations
TL;DR: In this paper, the authors compute asymptotic frequencies for the values taken by numerators of differences of consecutive Farey fractions with denominators restricted to lie in arithmetic progression. But their analysis is limited to the case where the difference is linear in the length of the last Farey fraction.
Abstract: As the conclusion of a line of investigation undertaken in two previous papers, we compute asymptotic frequencies for the values taken by numerators of differences of consecutive Farey fractions with denominators restricted to lie in arithmetic progression.
1 Jan 2011
TL;DR: Ramsey theory is the study of structure that must exist in a system, most typically after it has been partitioned as discussed by the authors, and it is the most commonly used theory in computer science.
Abstract: Ramsey theory is the study of structure that must exist in a system, most typically after it has been partitioned.
TL;DR: This paper proves the slightly stronger result that S(A,x)>=(2-@e)x for x>=x"0(@e,A).
TL;DR: In this paper, the notion of fractional density of a subset of the integers of zero density was defined, and a version of a theorem of Łaba and Pramanik on 3-term arithmetic progressions in subsets of the unit interval also holds for subsets with fractional densities whose characteristic functions have Fourier coefficients that decay sufficiently rapidly.
Abstract: Given a subset of the integers of zero density, we define the weaker notion of the fractional density of such a set. We show that a version of a theorem of Łaba and Pramanik on 3-term arithmetic progressions in subsets of the unit interval also holds for subsets of the integers with fractional density whose characteristic functions have Fourier coefficients that decay sufficiently rapidly.
TL;DR: This article studied the discrepancy of the distribution of arithmetic sequences in arithmetic progressions, and showed that the value of the first term of the progressions has a significant influence on the distribution.
Abstract: The goal of this article is to study the discrepancy of the distribution of arithmetic sequences in arithmetic progressions. We will fix a sequence $\A=\{\a(n)\}_{n\geq 1}$ of non-negative real numbers in a certain class of arithmetic sequences. For a fixed integer $a
eq 0$, we will be interested in the behaviour of $\A$ over the arithmetic progressions $a \bmod q$, on average over $q$. Our main result is that for certain sequences of arithmetic interest, the value of $a$ has a significant influence on this distribution, even after removing the first term of the progressions.
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TL;DR: In this article, a characterization of the divisors of 24 among all positive integers is given in terms of modular multiplication tables, based on Chinese remainder theorem, structure theory of units, Dirichlet's theorem on primes in an arithmetic progression, Bertrand-Chebyshev theorem, and results of Erdos and Ramanujan on the pi(x) function.
Abstract: What is an interesting number theoretic or a combinatorial characterization of the divisors of 24 amongst all positive integers? In this paper I will provide one characterization in terms of modular multiplication tables. This idea evolved interestingly from a question raised by a student in my elementary number theory class. I will give the characterization and then provide 5 proofs using various techniques: Chinese remainder theorem, structure theory of units, Dirichlet's theorem on primes in an arithmetic progression, Bertrand-Chebyshev theorem, and results of Erdos and Ramanujan on the pi(x) function.
TL;DR: An analogue of the Montgomery-Hooley asymptotic formula was established for the variance of the number of primes in arithmetic progressions, in which the moduli are restricted to the values of a polynomial.
Abstract: An analogue of the Montgomery-Hooley asymptotic formula is established for the variance of the number of primes in arithmetic progressions, in which the moduli are restricted to the values of a polynomial.
TL;DR: The problem of finding large arithmetic structures in S(A) (or generally in sumsets) is one of the most fundamental in combinatorial number theory as mentioned in this paper, especially for sufficiently dense sets A ⊆ {1,..., n} (|A| ≥ nα for some α > 1/2).
Abstract: the subsets sumset of A and let L(S) stand for the length of the longest arithmetic progression in S. The problem of finding large arithmetic structures in S(A) (or generally in sumsets) is one of the most fundamental in combinatorial number theory. It has been intensively studied, especially in the case of sufficiently dense sets A ⊆ {1, . . . , n} (|A| ≥ nα for some α > 1/2; see for example [1], [3], [4], [5]). A complete solution of this problem for sets of polynomial size was given recently by Szemerédi and Vu in [6], [7] and [8]. They proved, among other things, that if A ⊆ [n] and |A| d n1/d, where d ≥ 2 is a fixed integer, then
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TL;DR: In this paper, it was shown that all ordinates of zeros of a fixed Dirichlet function are linearly independent over the rationals, and an upper bound was established for the first member of the arithmetic progression that is not a zero of the function.
Abstract: Let $L(s,\chi)$ be a fixed Dirichlet $L$-function. Given a vertical arithmetic progression of $T$ points on the line $\Re(s)=1/2$, we show that $\gg T \log T$ of them are not zeros of $L(s,\chi)$. This result provides some theoretical evidence towards the conjecture that all ordinates of zeros of Dirichlet $L$-functions are linearly independent over the rationals. We also establish an upper bound (depending upon the progression) for the first member of the arithmetic progression that is not a zero of $L(s,\chi)$.
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TL;DR: In this paper, an upper bound for Mertens' function M(x,q,a) in arithmetic progression was shown for all moduli which are not too large.
Abstract: We consider Mertens' function M(x,q,a) in arithmetic progression, Assuming the generalized Riemann hypothesis (GRH), we show an upper bound that is uniform for all moduli which are not too large. For the proof, a former method of K. Soundararajan is extended to L-series.
TL;DR: The best known upper bound for w(k) is due to Gowers and is quite large as mentioned in this paper, for all k. The conjecture of van der Waerden and Gowers was later improved to 2k2 by Ron Graham.
Abstract: Several renowned open conjectures in combinatorics and number theory involve arithmetic progressions. Van der Waerden famously proved in 1927 that for each positive integer k there exists a least positive integer w(k) such that any 2-coloring of 1, . . . , w(k) produces a monochromatic k-term arithmetic progression. The best known upper bound for w(k) is due to Gowers and is quite large. Ron Graham [2] conjectures w(k) ≤ 2k2 , for all k. The
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TL;DR: In this article, the authors improved the estimate for the remainder term in the asymptotic formula concerning the circle problem in an arithmetic progression, and showed that it is possible to solve the problem in a linear fashion.
Abstract: In this paper we improve the estimate for the remainder term in the asymptotic formula concerning the circle problem in an arithmetic progression.
TL;DR: In this article, the product of k consecutive terms of a primitive arithmetic progression is never a perfect fifth power when 3 ⩽ k⩽ 54 and the product is an almost fifth power.
TL;DR: The lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence is improved, and lower bounds on the upper and lower densities of subsets of the positive integers that can be permuted to avoid 3- term and 4-term APs are derived.
TL;DR: A polynomial time algorithm is derived for a special case of the unbounded knapsack problem in which the item weights form an arithmetic sequence with running time O(n8), where n denotes the number of distinct items in the instance.
1 Jan 2011
TL;DR: In this paper, a smoothed version of the Polya-Vinogradov inequality (an explicit bound on character sums) and explicit estimates on the sum of primes are presented.
Abstract: Character sums make their appearance in many number theory problems: showing that there are infinitely many primes in any coprime arithmetic progression, estimating the least quadratic non-residue, bounding the least primitive root, finding the size of the least inert prime in a real quadratic field, etc. In this thesis, we find numerically explicit estimates for character sums and give applications to some of these questions. Granville, Mollin and Williams proved that the least inert prime q for a real quadratic field of discriminant D such that D > 3705, satisfies q ≤ √ D/2. Using a smoothed version of the Polya–Vinogradov inequality (an explicit bound on character sums) and explicit estimates on the sum of primes, we improve the bound on q to D for D > 1596. Let χ be a non-principal Dirichlet character mod p for a prime p. Using combinatorial methods, we improve an inequality of Burgess for the double sum
TL;DR: It is shown how known nonconstructive lower bound proofs based on the Lovasz Local Lemma can be made randomized-constructive using the recent algorithms of Moser and Tardos and a derandomization of Chandrasekaran, Goyal and Haeupler is used to transform these proofs into deterministic- constructive proofs.
Abstract: The van der Waerden number W(k,2) is the smallest integer n such that every 2-coloring of 1 to n has a monochromatic arithmetic progression of length k. The existence of such an n for any k is due to van der Waerden but known upper bounds on W(k,2) are enormous. Much effort was put into developing lower bounds on W(k,2). Most of these lower bound proofs employ the probabilistic method often in combination with the Lovasz Local Lemma. While these proofs show the existence of a 2-coloring that has no monochromatic arithmetic progression of length k they provide no efficient algorithm to find such a coloring. These kind of proofs are often informally called nonconstructive in contrast to constructive proofs that provide an efficient algorithm. This paper clarifies these notions and gives definitions for deterministic- and randomized-constructive proofs as different types of constructive proofs. We then survey the literature on lower bounds on W(k,2) in this light. We show how known nonconstructive lower bound proofs based on the Lovasz Local Lemma can be made randomized-constructive using the recent algorithms of Moser and Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to transform these proofs into deterministic-constructive proofs. We provide greatly simplified and fully self-contained proofs and descriptions for these algorithms.
TL;DR: Garcia-Selfa and Tornero as mentioned in this paper showed that 4319 is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over ℚ.
Abstract: A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (xi, yσ(i)), where and are arithmetic progressions and σ is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over ℚ. We show that 4319 is such a bound for curves over ℝ. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the "crossing inequality" gives a lower bound. Together these bound the length of an s.a.p. on the curve. We also extend this method to bound the k for which a real algebraic curve can contain k points from a k × k grid. Lastly, these results are extended to complex algebraic curves.
28 Mar 2011
TL;DR: In 2004, Ben Green and Terry Tao proved that, for every k, there are infinitely many length-k arithmetic progressions made entirely of prime numbers.
Abstract: In 2004, Ben Green and Terry Tao [6] proved that, for every k, there are infinitely many length-k arithmetic progressions made entirely of prime numbers. This settled a very long-standing open question in number theory that had been open even for the k = 4 case.