共查询到20条相似文献,搜索用时 46 毫秒
1.
Artūras Dubickas 《Monatshefte für Mathematik》2004,141(2):119-126
We investigate which algebraic numbers can be Mahler measures. Adler and Marcus showed that these must be Perron numbers. We prove that certain integer multiples of every Perron number are Mahler measures. The results of Boyd give some necessary conditions on Perron number to be a measure. These do not include reciprocal algebraic integers, so it would be of interest to find one which is not a Mahler measure. We prove a result in this direction. Finally, we show that for every non-negative integer k there is a cubic algebraic integer having norm 2 such that precisely the kth iteration of its Mahler measure is an integer. 相似文献
2.
Pete L. Clark 《manuscripta mathematica》2007,124(4):411-426
Fix a non-negative integer g and a positive integer I dividing 2g − 2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C
/K
of genus g and index I. This is obtained via a systematic analysis of local points on arithmetic surfaces with semistable reduction. Applications
are discussed to the corresponding problem over number fields. 相似文献
3.
Xue Gong Sun 《数学学报(英文版)》2010,26(1):155-160
Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2^k + p, where k is a positive integer and p is an odd prime. ErdSs ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2^k + p, and give a quantitative form of Romanoff's theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdos problem is affirmative. 相似文献
4.
Robert Hochberg 《Combinatorica》2006,26(1):47-64
We prove that the class of homogeneous quasi-arithmetic progressions has unbounded discrepancy. That is, we show that given any 2-coloring of the natural numbers and any positive integer D, one can find a real number α≥1 and a set of natural numbers of the form {0, [α], [2α], [3α], . . . , [kα]} so that one color appears at least D times more than the other color. This was already proved by Beck in 1983, but the proof given here is somewhat simpler and
gives a better bound on the discrepancy. 相似文献
5.
We study the maximum size and the structure of sets of natural numbers which contain no solution of one or more linear equations. Thus, for every natural i and k?2, we find the minimum α=α(i,k) such that if the upper density of a strongly k-sum-free set is at least α, then A is contained in a maximal strongly k-sum-free set which is a union of at most i arithmetic progressions. We also determine the maximum density of sets of natural numbers without solutions to the equation x=y+az, where a is a fixed integer. 相似文献
6.
Integral solutions toy
2=x
3+k, where either thex's or they's, or both, are in arithmetic progression are studied. When both thex's and they's are in arithmetic progression, then this situation is completely solved. One set of solutions where they's formed an arithmetic progression of length 4 had already been constructed. In this paper, we construct infinitely many
sets of solutions where there are 4x's in arithmetic progression and we disprove Mohanty's Conjecture [8] by constructing infinitely many sets of solutions where
there are 4, 5 and 6y's in arithmetic progression. 相似文献
7.
Theoretical and empirical convergence results for additive congruential random number generators 总被引:1,自引:0,他引:1
Roy S. Wikramaratna 《Journal of Computational and Applied Mathematics》2010,233(9):2302-151
Additive Congruential Random Number (ACORN) generators represent an approach to generating uniformly distributed pseudo-random numbers that is straightforward to implement efficiently for arbitrarily large order and modulus; if it is implemented using integer arithmetic, it becomes possible to generate identical sequences on any machine.This paper briefly reviews existing results concerning ACORN generators and relevant theory concerning sequences that are well distributed mod 1 in k dimensions. It then demonstrates some new theoretical results for ACORN generators implemented in integer arithmetic with modulus M=2μ showing that they are a family of generators that converge (in a sense that is defined in the paper) to being well distributed mod 1 in k dimensions, as μ=log2M tends to infinity. By increasing k, it is possible to increase without limit the number of dimensions in which the resulting sequences approximate to well distributed.The paper concludes by applying the standard TestU01 test suite to ACORN generators for selected values of the modulus (between 260 and 2150), the order (between 4 and 30) and various odd seed values. On the basis of these and earlier results, it is recommended that an order of at least 9 be used together with an odd seed and modulus equal to 230p, for a small integer value of p. While a choice of p=2 should be adequate for most typical applications, increasing p to 3 or 4 gives a sequence that will consistently pass all the tests in the TestU01 test suite, giving additional confidence in more demanding applications.The results demonstrate that the ACORN generators are a reliable source of uniformly distributed pseudo-random numbers, and that in practice (as suggested by the theoretical convergence results) the quality of the ACORN sequences increases with increasing modulus and order. 相似文献
8.
We consider the adic realization of the Morse transformation on the additive group of integer dyadic numbers. We discuss the
arithmetic properties of this action. Then we extend this action to an action of the group of rational dyadic numbers on the
solenoid. Bibliography: 14 titles.
To the memory of Alexander Livshits
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 70–90. 相似文献
9.
Keshav Mukunda 《Journal of Number Theory》2006,117(1):106-121
A Pisot number is a real algebraic integer, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is a Littlewood polynomial, one with {+1,-1}-coefficients, and shows that they form an increasing sequence with limit 2. It is known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Littlewood polynomials. Finally, we prove that every reciprocal Littlewood polynomial of odd degree n?3 has at least three unimodular roots. 相似文献
10.
R. T. Gregory 《BIT Numerical Mathematics》1981,21(2):194-202
A method is described for doing error-free computation when the operands are rational numbers. A rational operanda/b is mapped onto the integer ¦a·b
–1¦
p
and the arithmetic is performed inGF(p). A method is given for taking an integer result and finding its rational equivalent (the one which corresponds to the correct rational result). 相似文献
11.
Toufik Zaimi 《Archiv der Mathematik》2006,87(2):124-128
Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α∈P the fractional parts of the numbers
, when n runs through the set of positive rational integers, are dense in the unit interval if and only if N≦ 2d − 4. We also show that for any α∈P the integer parts of the numbers αn are divisible by N for infinitely many n if and only if N≦ 2d − 3.
Received: 27 April 2005 相似文献
12.
Zaizhao Meng 《Journal of Number Theory》2006,121(1):45-66
In this paper we obtain an improved asymptotic formula on the frequency of k-free numbers with a given difference. We also give a new upper bound of Barban-Davenport-Halberstam type for the k-free numbers in arithmetic progressions. 相似文献
13.
Artūras Dubickas 《Archiv der Mathematik》2007,88(1):29-34
Let K be a number field. We prove that the set of Mahler measures M(α), where α runs over every element of K, modulo 1 is everywhere dense in [0, 1], except when
or
, where D is a positive integer. In the proof, we use a certain sequence of shifted Pisot numbers (or complex Pisot numbers) in K and show that the corresponding sequence of their Mahler measures modulo 1 is uniformly distributed in [0, 1].
Received: 24 March 2006 相似文献
14.
Squares of Primes and Powers of 2 总被引:5,自引:0,他引:5
As an extension of the Linnik-Gallagher results on the “almost Goldbach” problem, we prove, among other things, that there
exists a positive integer k
0 such that every large even integer is a sum of four squares of primes and k
0 powers of 2.
(Received 7 September 1998; in revised form 3 May 1999) 相似文献
15.
Given a sequence B of relatively prime positive integers with the sum of inverses finite, we investigate the problem of finding B-free numbers in short arithmetic progressions. 相似文献
16.
For any square-free positive integer m, let H(m) be the class-number of the field , where ζm is a primitive m-th root of unity. We show that if m = {3(8 g + 5)}2 ? 2 is a square-free integer, where g is a positive integer, then H(4 m) > 1. Similar result holds for a square-free integer m = {3(8 g +7)}2 ?2, where g is a positive integer. We also show that n|H(4 m) for certain positive integers m and n. 相似文献
17.
Summary In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn 0 is a non-negative integer, then thenth triangular number isT
n
=n(n + 1)/2. Letk be a positive integer. We denote by
k
(n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculate
k
(n). The case wherek = 24 is particularly interesting. It turns out that, ifn 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 212(212 – 1)
24(n – 3). Furthermore the formula for
24(n) involves the Ramanujan(n)-function. As a consequence, we get elementary congruences for(n). In a similar vein, whenp is a prime, we demonstrate
24(p
k
– 3) as a Dirichlet convolution of
11(n) and(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres. 相似文献
18.
Abdelmejid Bayad 《Journal of Number Theory》2011,131(6):1020-1036
We introduce and investigate generalized poly-Bernoulli numbers and polynomials. We state and prove several properties satisfied by these polynomials. The generalized poly-Bernoulli numbers are algebraic numbers. We introduce and study the Arakawa-Kaneko L-functions. The non-positive integer values of the complex variable s of these L-functions are expressed rationally in terms of generalized poly-Bernoulli numbers and polynomials. Furthermore, we prove difference and Raabe?s type formulae for these L-functions. 相似文献
19.
Dimitrios Poulakis 《Monatshefte für Mathematik》1994,118(1-2):111-143
LetC: F(X, Y)=0 be an algebraic curve of genus 1, over a number fieldK. In this work we construct a modelG(Z,W)=0 of the curveC, over a fixed number fieldL with
, having the following property: ifx, y are algebraic integers ofK withF(x, y)=0, thenz=Z(x, y), w=W(x, y) are algebraic integers ofL withG(z, w)=0. Also, the total degree and the height of the polynomialG are bounded. As an application of this result, we give a reduction of the problem to determine effectively the integer points on a curve of genus 2, over a number field, to the problem to determine effectively the integer solutions of an equation of degree 4, over a number field. Also we consider a family of curvesF(X, Y)=0, defined over a number fieldK, which are cyclic coverings ofP
1 and we calculate, using our previous results, an explicit upper bound for the height of the integer points ofF(X, Y)=0 overK. 相似文献
20.
In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p2=x2+8(y2+z2) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2Tm(m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4. 相似文献