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Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.


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Journal ArticleDOI
01 Jan 2020
TL;DR: In this article, the authors studied upper bounds for cyclotomic numbers of order $e$ over the finite field Ω(Ω 1, √ n) of order n.
Abstract: Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $\mathbb{F}_q$. A general result of our study is that $(a,b)\leq 3$ for all $a,b \in \mathbb{Z}$ if $p> (\sqrt{14})^{k/ord_k(p)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0,0), (0,a), (a,0), (a,a)$ and $(a,b)$, where $a eq b$ and $a,b \in \{1,\dots,e-1\}$. The main idea we use is to transform equations over $\mathbb{F}_q$ into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.

5 citations

Patent
Frank W. Bennett1
29 Mar 1996
TL;DR: In this paper, a ripple through divider of a dividend by a constant is obtained by cascading a plurality of partial quotient tables, each table incorporates the same divisor, and each table need not appear as an input.
Abstract: A ripple through divider of a dividend by a constant is obtained by cascading a plurality of partial quotient tables. Each table incorporates the same divisor, so the divisor need not appear as an input. In one binary integer implementation for an n bit dividend that dividend is represented as n+1 bits having an MSB of 0. If the binary divisor is of k bits, then the most significant k+1 bits are applied to an input of a first partial quotient table. It produces one bit of fractionary quotient that becomes the MSB of the final quotient, and k bits of fractionary remainder. That fractionary remainder is combined as MSB's with an LSB that is the next and most significant unused dividend bit. This forms k+1 inputs to a second partial quotient table. It in turn produces a partial quotient bit that becomes the second most significant final quotient bit, and k-many more fractionary remainder bits. The cascading continues with additional stages of partial quotient tables until all dividend bits have been used. At that level the final quotient is available and the last partial remainder bits are indeed the actual final remainder bits. The partial quotient tables may be look-up tables implemented as ROM's or they may be constructed of discrete gating.

5 citations

Journal ArticleDOI
TL;DR: In this paper, a lower bound for σ2 is derived in terms of the excesses of the classes in the subgroup of the subfield of degree k ≥ 2q, where k is the mean value of the number Фj.
Abstract: Letp be a prime number ≡ 3 mod 4,G p the unit group of ℤ/pℤ, andg a generator ofG p. Letq be an odd divisor ofp - 1 andG p 2q = {t 2q;t ∈G pthe subgroup of index2q inG p. The groupG p 2 / p 2q consists of the classes $$\bar g^{2j} $$ ,j = 0,...,q – 1. In this paper we study the ’excesses’ of the classes $$\bar g^{2j} $$ in {l,...,(p–l)/2}, i.e., the numbers $$\Phi _j = \left| {\left\{ {k;1 \leqslant k \leqslant \left( {p - 1} \right)/2,\bar k \in \bar g^{2j} } \right\}} \right| - \left| {\left\{ {k;\left( {p - 1} \right)/2 \leqslant k \leqslant p - 1,\bar k \in \bar g^{2j} } \right\}} \right|$$ ,j = 0.....q — 1. First we express therelative class number h 2q of the subfieldK 2q⫃ ℚ(e2#x03C0;i/p ) of degree [K 2q: ℚ] =2q in terms of these excesses. We use this formula to establish certaincongruences for the Фj. E.g., ifq ∈ {3,5,11}, each number Фj is congruent modulo 4 to each other iff 2 dividesh - . Finally we study thevariance of the excesses, i.e., the number $$\sigma ^2 = ((\Phi _0 - \hat \Phi )^2 + \ldots + (\Phi _{q - 1} - \hat \Phi )^2 )/(q - 1)$$ , where $$\hat \Phi $$ is the mean value of the numbers Фj. We obtain an explicit lower bound for σ2 in terms ofh - /h - . Moreover, we show that log σ2 is asymptotically equal to 21og(h - h - )/(q - 1) forp→∞. Three tables illustrate the results.

5 citations

Patent
20 Jan 1982
TL;DR: In this article, the overflow condition of the quotient depends upon codes of the dividend and the divisor DR, and the overflow is detected by all figure zero of a lower rank n-1 figure of the DD, together with a result of a deciding circuit 2.
Abstract: PURPOSE:To immediately detect whether the quotient overflows or not, by combining codes of a dividend and a divisor, and directly comparing the dividend and the divisor, in case of a division for obtaining the quotient of (n) figures by a dividend of 2n figures and a divisor of (n) figures. CONSTITUTION:Numerical values are inserted into a register R1 and R2 for storing a dividend DD, and a register R3 for storing a divisor DR, and a result of division is stored in the register R2. Since an overflow condition of this quotient depends upon codes of the dividend DD and the divisor DR, gates 5-7, 8-10 are controlled by combination of the codes S, the divisor DR or its complement 3 is compared with higher rank figures 2-1-2n-2 of the dividend DD or its logical negation 4 by a comparator 1, and the overflow is detected by all figure zero of a lower rank n-1 figure of the dividend DD, together with a result of a deciding circuit 2. Its combination logic is executed by gates 13-22.

5 citations

Journal ArticleDOI
TL;DR: The average order of the arithmetic functions d(n) (the number of divisors of n E N) and r(n), i.e., the number of ways to write n as a sum of two integer squares, is a classical problem in analytic number theory as discussed by the authors.
Abstract: The study of the "average order" of the arithmetic functions d(n) (the number of divisors of n E N) and r(n) (the number of ways to write n as a sum of two integer squares) is a classical problem in analytic number theory: one usually considers the summatory functions ~n<_x d(n) and ~n<_x r(n) (Dirichlet's divisor problem and Gauss' circle problem, resp.; see the textbooks of FRICK~R [3] and KRXa'ZEL [9] for enlightening historical surveys and Iwaymc and Mozzocm [7], HUXLEY [6], and MOLLER and NOWAK [11] for the sharpest results known to date.) It seems to be a natural idea to shed some further light on this "average order" question by investigating sums of reciprocals, i.e. ~,_

5 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20222
2021157
2020172
2019127
2018120
2017140