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| In mathematics, an '''elliptic divisibility sequence (EDS)''' is a sequence of integers satisfying a nonlinear recursion relation arising from [[division polynomial]]s on [[elliptic curve]]s. EDS were first defined, and their arithmetic properties studied, by [[Morgan Ward]]<ref name="Ward">Morgan Ward, Memoir on elliptic divisibility sequences, ''Amer. J. Math.'' '''70''' (1948), 31–74.</ref>
| | By investing in a premium Word - Press theme, you're investing in the future of your website. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br> |
| in the 1940s. They attracted only sporadic attention until around 2000, when EDS were taken up as a class of nonlinear recurrences that are more amenable to analysis than most such sequences. This tractability is due primarily to the close connection between EDS and elliptic curves. In addition to the intrinsic interest that EDS have within number theory, EDS have applications to other areas of mathematics including [[logic]] and [[cryptography]]. | |
|
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|
| == Definition ==
| | Should you loved this informative article and you wish to receive more info relating to [http://aorta.in/WordpressDropboxBackup925594 wordpress backup plugin] please visit our own web site. Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. We also help to integrate various plug-ins to expand the functionalities of the web application. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. Setting Up Your Business Online Using Free Wordpress Websites. Enough automated blog posts plus a system keeps you and your clients happy. For any web design and development assignment, this is definitely one of the key concerns, specifically for online retail outlets as well as e-commerce websites. <br><br>Word - Press installation is very easy and hassle free. Find more information about Design To Wordpress here. To do this, you should link your posts to other relevant posts that you've created. So, we have to add our social media sharing buttons in website. Word - Press offers constant updated services and products, that too, absolutely free of cost. <br><br>Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. An ease of use which pertains to both internet site back-end and front-end users alike. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. Page speed is an important factor in ranking, especially with Google. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
| A (nondegenerate) ''elliptic divisibility sequence'' (EDS) is a sequence of integers {{math|(<var>W<sub>n</sub></var>)<sub><var>n</var> ≥ 1</sub>}}
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| defined recursively by four initial values
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| {{math|<var>W</var><sub>1</sub>}}, {{math|<var>W</var><sub>2</sub>}}, {{math|<var>W</var><sub>3</sub>}}, {{math|<var>W</var><sub>4</sub>}},
| |
| with {{math|<var>W</var><sub>1</sub><var>W</var><sub>2</sub><var>W</var><sub>3</sub>}} ≠ 0 and with subsequent values determined by the formulas
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| | |
| :<math>
| |
| \begin{align}
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| W_{2n+1}W_1^3 &= W_{n+2}W_n^3 - W_{n+1}^3W_{n-1},\qquad n \ge 2, \\
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| W_{2n}W_2W_1^2 &= W_{n+2}W_n W_{n-1}^2 - W_n W_{n-2}W_{n+1}^2,\qquad n\ge 3,\\
| |
| \end{align}
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| </math>
| |
| | |
| It can be shown that if {{math|<var>W</var><sub>1</sub>}} divides each of {{math|<var>W</var><sub>2</sub>}}, {{math|<var>W</var><sub>3</sub>}}, {{math|<var>W</var><sub>4</sub>}} and if further {{math|<var>W</var><sub>2</sub>}} divides {{math|<var>W</var><sub>4</sub>}}, then every term {{math|<var>W<sub>n</sub></var>}} in the sequence is an integer.
| |
| | |
| == Divisibility property ==
| |
| An EDS is a [[divisibility sequence]] in the sense that
| |
| :<math> | |
| m \mid n \Longrightarrow W_m \mid W_n.
| |
| </math>
| |
| In particular, every term in an EDS is divisible by {{math|<var>W</var><sub>1</sub>}}, so
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| EDS are frequently ''normalized'' to have {{math|<var>W</var><sub>1</sub>}} = 1 by dividing every term by the initial term.
| |
| | |
| Any three integers {{math|<var>b</var>}}, {{math|<var>c</var>}}, {{math|<var>d</var>}}
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| with {{math|<var>d</var>}} divisible by {{math|<var>b</var>}} lead to a normalized EDS on setting
| |
| :<math>
| |
| W_1 = 1,\quad W_2 = b,\quad W_3 = c,\quad W_4 = d.
| |
| </math>
| |
| It is not obvious, but can be proven, that the condition {{math|<var>b</var>}} | {{math|<var>d</var>}} suffices to ensure that every term
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| in the sequence is an integer. | |
| | |
| == General recursion ==
| |
| A fundamental property of elliptic divisibility sequences
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| is that they satisfy the general recursion relation
| |
| :<math>
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| W_{n+m}W_{n-m}W_r^2 = W_{n+r}W_{n-r}W_m^2 - W_{m+r}W_{m-r}W_n^2
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| \quad\text{for all}\quad n > m > r.
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| </math>
| |
| (This formula is often applied with {{math|<var>r</var>}} = 1 and {{math|<var>W</var><sub>1</sub>}} = 1.)
| |
| | |
| == Nonsingular EDS ==
| |
| The ''discriminant'' of a normalized EDS is the quantity
| |
| :<math>
| |
| \Delta =
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| W_4W_2^{15} - W_3^3W_2^{12} + 3W_4^2W_2^{10} - 20W_4W_3^3W_2^7 +
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| 3W_4^3W_2^5 + 16W_3^6W_2^4 + 8W_4^2W_3^3W_2^2 + W_4^4.
| |
| </math>
| |
| An EDS is ''nonsingular'' if its discriminant is nonzero.
| |
| | |
| == Examples ==
| |
| A simple example of an EDS is the sequence of natural numbers 1, 2, 3,… . Another interesting example is the sequence 1, 3, 8, 21, 55, 144, 377, 987,… consisting of every other term in the [[Fibonacci sequence]], starting with the second term. However, both of these sequences satisfy a linear recurrence and both are singular EDS. An example of a nonsingular EDS is
| |
| :<math>
| |
| \begin{align}
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| &1,\, 1,\, -1,\, 1,\, 2,\, -1,\, -3,\, -5,\, 7,\, -4,\, -23,\,
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| 29,\, 59,\, 129,\\
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| &-314,\, -65,\, 1529,\, -3689,\, -8209,\, -16264,\dots.\\
| |
| \end{align}
| |
| </math>
| |
| | |
| == Periodicity of EDS ==
| |
| A sequence {{math|(<var>A<sub>n</sub></var>)<sub><var>n</var> ≥ 1</sub>}} is said to be ''periodic''
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| if there is a number {{math|<var>N</var> ≥ 1}} so
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| that {{math|<var>A<sub>n+N</sub></var>}} = {{math|<var>A<sub>n</sub></var>}} for every {{math|<var>n</var>}} ≥ 1.
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| If a nondegenerate EDS {{math|(<var>W<sub>n</sub></var>)<sub><var>n</var> ≥ 1</sub>}}
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| is periodic, then one of its terms vanishes. The smallest {{math|<var>r</var>}} ≥ 1 with {{math|<var>W<sub>r</sub></var>}} = 0 is called the ''rank of apparition'' of the EDS. A deep theorem of Mazur<ref name="Mazur">
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| B. Mazur.
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| Modular curves and the Eisenstein ideal,
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| ''Inst. Hautes Études Sci. Publ. Math.'' 47:33–186, 1977.
| |
| </ref>
| |
| implies that if the rank of apparition of an EDS is finite, then it satisfies {{math|<var>r</var>}} ≤ 10 or {{math|<var>r</var>}} = 12.
| |
| | |
| == Elliptic curves and points associated to EDS ==
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| Ward proves that associated to any nonsingular EDS ({{math|<var>W<sub>n</sub></var>}})
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| is an elliptic curve {{math|<var>E</var>}}/'''Q''' and a point
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| {{math|<var>P</var>}} ε {{math|<var>E</var>}}('''Q''') such that
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| :<math>
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| W_n = \psi_n(P)\qquad\text{for all}~n \ge 1.
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| </math>
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| Here ψ{{math|<var><sub>n</sub></var>}} is the
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| [[division polynomial|{{math|<var>n</var>}} division polynomial]]
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| of {{math|<var>E</var>}}; the roots of ψ{{math|<var><sub>n</sub></var>}} are the | |
| nonzero points of order {{math|<var>n</var>}} on {{math|<var>E</var>}}. There is
| |
| a complicated formula<ref name="SilvermanStephens"> | |
| This formula is due to Ward. See the appendix to J. H. Silverman and N. Stephens.
| |
| The sign of an elliptic divisibility sequence. ''J. Ramanujan Math. Soc.'', 21(1):1–17, 2006.
| |
| </ref>
| |
| for {{math|<var>E</var>}} and {{math|<var>P</var>}} in terms of {{math|<var>W<sub>1</sub></var>}}, {{math|<var>W<sub>2</sub></var>}}, {{math|<var>W<sub>3</sub></var>}}, and {{math|<var>W<sub>4</sub></var>}}.
| |
| | |
| There is an alternative definition of EDS that directly uses elliptic curves and yields a sequence which, up to sign, almost satisfies the EDS recursion. This definition starts with an elliptic curve {{math|<var>E</var>}}/'''Q''' given by a Weierstrass equation and a nontorsion point {{math|<var>P</var>}} ε {{math|<var>E</var>}}('''Q'''). One writes the {{math|<var>x</var>}}-coordinates of the multiples of {{math|<var>P</var>}} as
| |
| :<math>
| |
| x(nP) = \frac{A_n}{D_n^2} \quad \text{with}~\gcd(A_n,D_n)=1~\text{and}~D_n \ge 1.
| |
| </math>
| |
| Then the sequence ({{math|<var>D<sub>n</sub></var>}}) is also called an '''elliptic divisibility sequence'''. It is a divisibility sequence, and there exists an integer {{math|<var>k</var>}} so that the subsequence ( ±{{math|<var>D<sub>nk</sub></var>}} )<sub>{{math|<var>n</var>}} ≥ 1</sub> (with an appropriate choice of signs) is an EDS in the earlier sense.
| |
| | |
| == Growth of EDS ==
| |
| Let {{math|(<var>W<sub>n</sub></var>)<sub><var>n</var> ≥ 1</sub>}} be a nonsingular EDS
| |
| that is not periodic. Then the sequence grows quadratic exponentially in the sense that there is
| |
| a positive constant {{math|<var>h</var>}} such that
| |
| :<math>
| |
| \lim_{n\to\infty} \frac{\log |W_n|}{n^2} = h > 0.
| |
| </math> | |
| The number {{math|<var>h</var>}} is the [[canonical height]] of the point on
| |
| the elliptic curve associated to the EDS.
| |
| | |
| == Primes and primitive divisors in EDS ==
| |
| It is conjectured that a nonsingular EDS contains only finitely many
| |
| primes<ref name="Einsiedler">
| |
| M. Einsiedler, G. Everest, and T. Ward. Primes in elliptic divisibility sequences.
| |
| ''LMS J. Comput. Math.'', 4:1–13 (electronic), 2001.
| |
| </ref>
| |
| However, all but finitely many terms in a nonsingular EDS admit a primitive prime
| |
| divisor.<ref name="Silverman">
| |
| J. H. Silverman. Wieferich's criterion and the ''abc''-conjecture.
| |
| ''J. Number Theory'', 30(2):226–237, 1988.
| |
| </ref>
| |
| Thus for all but finitely many {{math|<var>n</var>}},
| |
| there is a prime {{math|<var>p</var>}} such that {{math|<var>p</var>}} divides {{math|<var>W<sub>n</sub></var>}}, but {{math|<var>p</var>}} does not divide {{math|<var>W<sub>m</sub></var>}} for all {{math|<var>m</var>}} < {{math|<var>n</var>}}. This statement is an analogue of [[Zsigmondy's theorem]].
| |
| | |
| == EDS over finite fields ==
| |
| An EDS over a finite field '''F'''<sub>{{math|<var>q</var>}}</sub>, or more generally over any field, is a sequence of elements of that field satisfying the EDS recursion. An EDS over a finite field is always periodic, and thus has a rank of apparition {{math|<var>r</var>}}. The period of an EDS over '''F'''<sub>{{math|<var>q</var>}}</sub> then has the form {{math|<var>rt</var>}}, where {{math|<var>r</var>}} and {{math|<var>t</var>}} satisfy
| |
| :<math>
| |
| r \le \left(\sqrt q+1\right)^2 \quad\text{and}\quad t \mid q-1.
| |
| </math>
| |
| More precisely, there are elements {{math|<var>A</var>}} and {{math|<var>B</var>}} in '''F'''<sub>{{math|<var>q</var>}}</sub><sup>*</sup> such that
| |
| :<math>
| |
| W_{ri+j} = W_j\cdot A^{ij} \cdot B^{j^2}
| |
| \quad\text{for all}~i \ge 0~\text{and all}~j \ge 1.
| |
| </math>
| |
| The values of {{math|<var>A</var>}} and {{math|<var>B</var>}} are related to the
| |
| [[Tate pairing]] of the point on the associated elliptic curve.
| |
| | |
| == Applications of EDS ==
| |
| [[Bjorn Poonen]]<ref>
| |
| B. Poonen. Using elliptic curves of rank one towards the undecidability of
| |
| Hilbert's tenth problem over rings of algebraic integers.
| |
| In ''Algorithmic number theory (Sydney, 2002)'', volume 2369 of
| |
| ''Lecture Notes in Comput. Sci.'', pages 33–42. Springer, Berlin, 2002.
| |
| </ref>
| |
| has applied EDS to logic. He uses the existence of primitive divisors in EDS on elliptic curves of rank one to prove the undecidability of [[Hilbert's tenth problem]] over certain rings of integers.
| |
| | |
| [[Katherine Stange]]<ref name="Stange">
| |
| K. Stange. The Tate pairing via elliptic nets.
| |
| In ''Pairing-Based Cryptography (Tokyo, 2007)'', volume 4575 of
| |
| ''Lecture Notes in Comput. Sci.'' Springer, Berlin, 2007.
| |
| </ref>
| |
| has applied EDS and their higher rank generalizations called [[elliptic nets]]
| |
| to cryptography. She shows how EDS can be used to compute the value
| |
| of the [[Weil pairing|Weil and Tate pairings]] on elliptic curves over finite
| |
| fields. These pairings have numerous applications in [[pairing-based cryptography]].
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| == Further material ==
| |
| | |
| * G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward. ''Recurrence sequences'', volume 104 of ''Mathematical Surveys and Monographs''. American Mathematical Society, Providence, RI, 2003. ISBN 0-8218-3387-1. (Chapter 10 is on EDS.)
| |
| * R. Shipsey. [http://homepages.gold.ac.uk/rachel/rachthesis.ps.gz ''Elliptic divisibility sequences'']. PhD thesis, Goldsmith's College (University of London), 2000.
| |
| * K. Stange. ''Elliptic nets''. PhD thesis, Brown University, 2008.
| |
| * C. Swart. [http://www.isg.rhul.ac.uk/files/alumni/thesis/swart_c.pdf ''Sequences related to elliptic curves'']. PhD thesis, Royal Holloway (University of London), 2003.
| |
| | |
| == External links ==
| |
| * [http://www.mth.uea.ac.uk/%7Eh090/EDS.html Graham Everest's EDS web page.]
| |
| * [http://www.mth.uea.ac.uk/~h090/primeEDS.html Prime Values of Elliptic Divisibility Sequences.]
| |
| * [http://www.math.brown.edu/~jhs/Presentations/ICMSEDSLecture.pdf Lecture on ''p''-adic Properites of Elliptic Divisibility Sequences.]
| |
| | |
| [[Category:Number theory]]
| |
| [[Category:Integer sequences]]
| |
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