Talk:P-adic number

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infinite p-adic representations on an illustration only showing finite digits, but in his own proposed illustration he still must cut an infinite representation short, which is the very nature of these things. Even if the lay reader does not understand the whole image, the middle section is a very clear illustration of the topology of the p-adics. The SVG even tagged the balls, which are not clear in the article space, perhaps this is to be remedied somehow. The picture is striking and Pontryagin Duality gives a geometric association to a subject which too easily could appear overly formal to the lay reader. The color pallete is completely in line with those used in 3d complex plots, with argument represented by a value on a color wheel. These are often hailed as stunning visuals that get people intrigued in a number system initially beyond their intuition, even if the plots can stay largely mysterious. There is another illustration using 2-adic numbers. It's a good one, but 90-degree angles are some of the most recognized and can lead to some confusion as to groupings as the groupings can easily be inferred to be in groups of 4 (of course filtering the limit to powers of 4 changes nothing about the 2-adic integers, but that is not the most natural way of thinking about them). Hence I think 3 is a better choice.

5191:– this may be, but that's because those complex plots (what Wegert calls phase portraits) have an incredibly poorly chosen color scheme (which is not surprising: the color scheme was barely "chosen" at all, but rather was the easiest thing to implement for various mathematicians/programmers who had no background whatsoever in visual art, design, or vision science). The problems with it are that (1) the colors create visual artifacts that are not relevant to the data, (2) the colors hide visual features of the data that should be presented, (3) the color relationships give misleading visual impression of data relationships, and perhaps especially (4) the colors are so intense that they are visually distracting and unpleasant (perhaps even physically uncomfortable) to look at. One of my medium term goals is to take my painfully slow Matlab code that generates dramatically better phase portraits than the ones found around the web, and properly reimplement it in a browser to render using a GPU. Ricky Reusser implemented 5103:). If one stops at, say, the level 3 (27 nodes at this level), one may add some nodes at infinity such as -1, 1/2, -1/2, 5/7, ..., with dotted arrows toward their reduction modulo 27. I suggest to present the image horizontally with the root on the right. Probably, the level 3 would give a to complicate image. This can be solved by stopping at level 2 or considering 2-adic numbers. In any case, 2-adic numbers seem more suitable than 3-adic numbers, as allowing more information with less nodes. Also, readers are probably more accustomed with base 2 than with base 3. 5140: 4695: 307: 297: 276: 243: 5214: 388: 2090:-adic series of your normalized type. And it is possible to define arithmetics (addition, subtraction, multiplication, division) on these series of your normalized type and to show that this arithmetic coincides for rational numbers with the standard arithmetic ― very much similar to the teachers for decimal arithmetic at school. These definitions and proofs are on solid mathematical foundations and there is 203: 5155:* The 2-adic version is shown at the right. It has roughly the same formal content, but I agree with Qsdd that the 3-adic image is better. In part, this is because the 2-adic version makes the black points sparser (lower Hausdorff dimension), so the 2-adic numbers are less visually salient than the metric balls and the selected duals. The lead image for an article about the 234: 4667: 5166:* (Also, the simple fraction 1/2 is not a 2-adic integer, which is a shame. I think the elements −1/2 and 1/2 are the most visually stunning elements of the 3-adic integers in the 3-adic image. The 2-adic image shows −1/3 and 1/3, but the colored discs for those are much less comprehensible, in my opinion.) 2286:

Besides the unresolved issue about the „p-adic series” above, I think it is not a good sequence of explaining some math analysis when the series do run up to infinity without having defined the conditions for convergence. Many books introduce the possible valuations (absolute values) of the rational

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As for (4), that's more subjective. I could try dialing back the green a bit. Not too much, though, because the variation in lightness serves a mathematical purpose! It helps to communicate the mirror-image relationship between the pinwheel shapes on the right, especially −9 vs 9. The difference in

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If there's otherwise no image at all, I think this is too harsh a standard / too harsh an application of the standard. Do you have an idea for a more comprehensible image that would be meaningful to an audience at, say, high school level? If not, an image that flies over some readers heads is still

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I'm open to improving the image to have more explanatory value, although at the outset, I want to warn that there's only so much explanation that is possible in a lead image. The image description page has plenty of room for explanatory text, but the image itself doesn't, nor does the caption. It's

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is already confusing, as 10 is not prime. Moreover, the fact that 10 is not prime induces complications that makes the introduction harder to understand than the subject of the article, and leads to nonsensical assertions, such the definition of an "absolute value" that is explicitly said to not be

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The colors absolutely should not all have the same lightness (lightness contrast is the most visually salient for recognizing fine details, and a picture without lightness contrast looks entirely flat and becomes difficult for readers with normal color vision but nearly impossible for readers with

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why is it that wiki math articles have the worst intros, from the POV of readability by a general reader this article's intro is not pitched for a general reader; it, to use a technical term from higher math, sucks these are better but still not good I am just so pissed off that you math people in

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picks out a few representative 3-adic integers to label, namely 0, 9, 3, 1, −1/8, −1/2, 1/4, −1/4, 1/2, 1/8, −1, −3, and −9. Those labels are already explained in the image description page. The proposal is merely to embed the labels in the content of the image, as well. There is no need to worry

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I restored the image in a flight of passion before reading the talk page. I did not intend to edit war, I am sorry. It is one of my favorite mathematical illustrations. I don't think the criticisms of the image are strong enough to warrant its deletion. D.Lazard criticizes the image for labelling

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Thanks for the suggestions. I've added a link to the Quanta magazine article to the External links section. I'll look at possible ways to improve the intro, but many math topics are difficult to explain even for specialists. Also please add "--~~~~ to the end of your comments. This adds you

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One general request (not specific to this image): can people please choose colors that are less colorful than #00F, #0F0, #F00, etc.? If anyone needs help choosing a color scheme I am happy to offer more concrete advice based on whatever constraints are relevant to the particular image.

1750: 4711:. I created the image, so I'm biased! But I think there's a lot of value in having a lead image that is compelling and, dare I say, pretty. And I want to emphasize that the image isn't some flight-of-fancy original research on my part. It's just the solenoidal embedding of the 504:

I intended to remove this section again, but it appeared to me that the whole article has similar issues. Especially, I have not found any workable definition of p-adic integers and p-adic numbers. So, I'll add the lacking information before removing the present mess.

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but I don't think all of the details are quite right yet. Once that is sorted I'd like to try to replace as many of the phase portraits found around the world as possible with better ones that more appropriately and pleasantly depict the data they are trying to show.

1062:-adic series of D.Lazard type ? Since I would like to assume that you have something in mind with them, it would only be fair to your WP-readers that you tell them what it is. Just to make further speculations superfluous and the discussion simpler. ― 482:, confusing, and/or out of scope: As 10 is not prime, the "10-adic absolute value" that is considered here is not an absolute value, and the strange resulting properties do not help to understand the subject of the article". This has been reverted. 1439: 4846:

About your suggested modification of the image, I doubt that it will make the image clearer for the lay reader. Moreover, I do not understand how you intend to label nodes of a finite tree (this is the nature of the image) with infinite 3-adic

2340:-adic numbers were introduced for number theory, and most of their applications are in number theory and algebra. So, they must be described in a style adapted for number theorists and algebraists. This was not the case before my edits. 829:

Besides that, there is derived an equivalence relation which IMHO is far less important than the equivalence relation which is established by the sequences converging to 0, which means that equivalent sequences (or series) have the same

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Here's the thing about the naive RGB color scheme. In 99% of cases, I would agree with the framing of (1-3) as problems that should be avoided. Just to prove that I'm being sincere, see the illustration on the right, where I didn't use

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to talk about inverse limits nor about modular properties, but ―yes― carry management is needed. Btw, you did not yet really show how solid your mathematical foundations are: for me and up to now, the usefulness of the unnormalized

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I took a shot at a more accessible first sentence, but folks should feel free to adapt or rewrite it for clarity/style. Can we find any existing materials aimed at laypeople which have a clear 1–2 sentence summary?

153: 4828:. Here, the image caption involves advanced knowledge of the theory of topological groups, which are not supposed to be known by readers; even for people who know the involved concepts, the relationship between 2484: 1569: 4634:-adic series". This is only after publishing it that I saw that a section existed already with this title, in a style that resemble to mine. In fact, it is myself who partially wrote the section two year ago. 4836:-adic numbers, and it is only "decorative" for most readers. Moreover, it is very confusing, as suggesting that the group structure (additive or multiplicative?) is more fundamental than the field structure. 4337: 4232: 4127: 1562: 4773:

Such labels would have the nice benefit of reinforcing the lead paragraph's discussion of digit expansions. They would make the whole image larger, so to avoid losing detail, I would want to increase the

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lightness between green and blue makes those characters visually "turn" in opposite directions. If all the colors had the same lightness, I worry that the whole thing would degrade into visual noise.

1506: 1231: 5101: 3494: 910:-adic numbers and a description of their main properties that is accessible to the largest possible audience. I am working on this, and this is for this reason that I have added the section " 619: 2852:. And it has the additional (in my opinion lesser) problem of 10 not being a prime. Finally, it is indeed a question whether starting from decimal standard has to be given up in principle. ― 2358:

If you identify math analysis with real analysis you may be right. But if analysis starts with infinitesimal and infinity then you are wrong, then it is analysis already when you talk about

1272: 363: 3405: 5318: 1786:-adic series is implicit in their positional notation, and this positional notation is used in many introducing textbook. As Knowledge must be based on a solid mathematical foundation, 2052:). I'll detail this approach in a section entitled "Modular properties". This is important as this is widely used for fast integer and rational computation (exact linear algebra), and 3070: 4864:

better than nothing. It seems fair enough to call this image significant and relevant, not "primarily decorative", even if it is not immediately obvious to less technical readers. –

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x | x1 | x2 | x3 | x4 -------+----+----+----+---- 1 | ? | ? | ? | ? 2 | ? | ? | ? | ? 3 | ? | ? | ? | ? ... 8 | ? | ? | ? | ? ...

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But if ever there were a time to introduce visual artifacts that emphasize the special role of the cube roots of unity, the diagram of the 3-adic integers would be the time. It

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In the end, the end is not visible. So I'll let you time for your elaborations. We both know that the matter is/the matters are settled in principle ― maybe already since 1897.

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as the standard user is insinuated to assume. And it is absolutely not obvious and has to be proved explicitly that your equivalence relation comprises all rational numbers. ―

4427: 2835:, but how do you get there ? Since these algorithms are all well known and already contained in WP, I'm asking why you do not take them ― and present your phantasy instead ? 2747: 2691: 2620:-adic expansion of a rational number is still lacking. Also, the algorithms are only sketched, but a section detailing them is needed and lacking (even in the old version). 2538: 2395: 5152:

Thanks for the comments! I'm not as active on Knowledge as I used to be, so I might not be able to respond to every point of discussion. Just a couple of things to clarify:

3236: 2972: 2314:, and thus that no convergence has to be considered. In any case circular reasoning must be avoided, and it seems that this is what you are asking for: one cannot define a 194: 3122: 1279: 4951: 5030: 2333:-adic expansion" widely considered in the article, with the difference that writing the terms of the series as digits makes impossible to work with large prime numbers. 1966: 147: 5308: 4715:-adic integers. The description for the full-size image cites Chistyakov (1996), but you can find similar images in the dynamical systems literature even before that. 3159: 2927: 4377: 4020: 3977: 3782: 3548: 3521: 3432: 2568: 1933: 813: 646: 557: 3185: 3016: 2838:

I'm extremely eager to learn about your detailing the «only sketched» algorithms of addition, subtraction, multiplication, division (I do not see a sketch anywhere).

2791: 2325:-adic series" is not a standard term. This is the reason for introducing them by "in this article". However, the concept is not original research, as my "normalized 772: 4997: 2890: 5323: 4971: 4918: 1445:

A very simple thing for you could be to pinpoint to an operation (addition, subtraction, multiplication, division) and to related operands which are not closed. ―

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Up to now, I have not yet considered any math analysis. Everything that I have written is pure algebra. I have just edited the article for making clear that the

743: 4843:-adic numbers, such an image could be used to illustrate the section. If such a section is eventually added, the image must be in that section, not in the lead. 4637:

So, these two sections need to be merged, and I will do this soon. However, the old section contains some mistakes. Mainly that, with the chosen definition of

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an absolute value (moreover this caveat is in a footnote, and this makes the section even more confusing for readers who do not spent time to open footnotes).

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color vision deficiencies). However, the lightness used for each color should be deliberately chosen, not just left to the arbitrary primaries used in sRGB. –

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As it appears you have finished working on this article. So I insert some amendments which make clear that not everything is clear, especially this section „

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Btw, I threw out your remark on the size of the base, because you are far away from showing an example where a problem about the size of the digits shows up.

247: 1097:-adic series. So the question is not "What is «unnormalized p-adic series» good for", it is "Is there a reasonable way to define and describe operations on 5333: 1089:-adic series are not closed under series operations (addition, subtraction, multiplication, division, etc.). The series operations applied to normalized 353: 79: 3632: 1971: 5303: 2291:

rational numbers (and not only those with p-denominators) are among these convergent series and can if needs be written as infinite p-adic series. –

5313: 3827: 329: 887:, or general notions of topology (completion). Moreover the equivalence of these definitions is not mentioned. Also many properties that make 5328: 1745:{\displaystyle \sum _{i=0}^{\infty }5^{i}-\sum _{i=0}^{\infty }3\cdot 5^{i}=\sum _{i=0}^{\infty }(1-3)5^{i}=\sum _{i=0}^{\infty }(-2).5^{i}.} 85: 44: 5174:

about infinite digit expansions, since the selected elements are all rational numbers with repeating expansions, which I wrote down above.

4379:? It is non-obvious in the article. And, if the Gupta's definition is valid, how to relate the two representations? they are equivalent? 4394:

I agree that the article is awfully written. I have rewritten the lead for having a clear (and as elementary as possible) definition of

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series which run to infinity. And you certainly know that the difference between pure algebra and other math does not help you so much.

2422: 4832:-adic numbers and elements of the figure is far to be obvious. So, the image is definitively not an "illustrative aid to understand" 2193:-adic numbers as infinite sequences of digits use formal series implicitly and cannot be made formally correct without formal series. 4788:

Before I go to the trouble of putting such an SVG together, what do you think? Of course, other editors' opinions are also welcome.

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Pls do not misunderstand me: I do not want to defend the previous version. I fully agree with your goal: to provide a definition of

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it to be obvious where that happens. The usage of special colors is -- in this one, exceptional case -- a feature, not a bug.

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numbers first, then define a topology, then define limits and completion within, and then (in the case of p-adics) show that

1853: 1101:-adic numbers, without using unnormalized p-adic series and without entering into the technical details of carry management? 104: 20: 2103:-adic notation appears soon in the article and I have decided to wait, although your latest post does not help me very much. 168: 1123:-adic series ARE closed under addition, subtraction, multiplication, division. I am sure that you know this as well: Isn't 1461: 135: 74: 5192: 1756:

This is a 5-adic series, but not a normalized one. One must normalize it for getting the difference as a 5-adic numbers.

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As far as I can see the old version recurs to the well-known standard algorithms of finite partial sums and goes to the

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This is certainly a possibility, but done without any reference ― and completely new to me. In all books I know of, the

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Images must be significant and relevant in the topic's context, not primarily decorative. They are often an important

2200:-adic series is explicitly restricted to this article, it has been introduced for convenience of writing. It would be 431: 65: 3440: 565: 202: 185: 1236: 5265:

Oh interesting, I thought we were going to disagree about lightness contrast, but I'm glad to have been mistaken!

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meaningful that certain characters assign certain elements of the dual group to certain cube roots of unity. We

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an answer to my question: How do you find this order (after a mathematical operation) without any evaluation ? ―

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and you have to manage some carry. And I'm sure that you can't avoid it when you want to show some equivalence:

448: 410: 213: 4588: 3349: 2053: 975:-adic numbers and a description of their main properties that is accessible to the largest possible audience. ― 463: 129: 4719:

asking too much of the lead image to fully explain the topic to a general-audience reader with no background.

2848:(!) limits. But I agree: this is not really well elaborated, if you do not take the link to the very strange 2397:: as far as I can see the identification of these series with their normalizations can work only if you take 2099:-adic series of D.Lazard type is hot air including the arithmetics on them. I can agree that some positional 109: 3022: 489:

the subject of the article which is p-adic numbers. In the literature, these numbers are defined only when

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So I have tagged the section, and explain here whic this section must be removed. This section is aimed to

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From your post I have to assume that you don't like carry management. But also with your unnormalized

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In summary, something is "far more important" than anything else, which is to provide a definition of

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Your recourse to formal series is a problem especially when you admit coefficients with denominators

2296: 2292: 2269: 2265: 2167: 2163: 2114: 2110: 1450: 1446: 1434:{\displaystyle \sum _{i=k}^{n}a_{i}p^{i}-\sum _{i=k}^{n}b_{i}p^{i}=\sum _{i=k}^{n}(a_{i}-b_{i})p^{i}} 1126: 1067: 1063: 1022: 1018: 980: 976: 839: 835: 175: 5163:-adic numbers front and center, so that's an important consideration in favor of the 3-adic version. 1050:

In your edit dated 12 June 2021 08:45 you have added: «every rational number can be considered as a

233: 5257: 5200: 4884: 4868: 4823: 4808: 4775: 4602: 4584: 3984: 2220: 2208: 2159: 1153: 459: 161: 55: 2802:=−3/2 ? Then, what would be the subsequent division step ?? And the 3-adic expansion of 1/2 ?? It 2601:

Clearly some tweaks are still needed. I have fixed your objection about the numerator multiple of

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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I'm disappointed by the removal of the previous lead image (shown at right) in this August edit:

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Pls note that it has the additional advantage that its arithmetics is taught in school !! ―

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You do not elaborate true algorithms for addition, subtraction, multiplication, division.

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containing all rational numbers and all numbers, you are talking about (the unnormalized

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I would like to emphasize my remark 2: There exist people (I'm not among them) who find

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of a rational number, but, to be found, it requires a careful reading of the article.

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As it appears in total: you do not have a book where you are taking your texts from.

2318:-adic number as a limit when the space in which the limit occurs is not yet defined. 4893: 2584: 2489: 2216: 2212: 2201: 2182: 479: 4579:

That example still resembles positional notation, to any programmer familiar with

3720:{\displaystyle -{\frac {1}{3}}=2+3\cdot 5+1\cdot 5^{2}+{\frac {-1}{3}}\cdot 5^{3}} 2037:{\displaystyle \mathbb {Z} /p^{i+1}\mathbb {Z} \to \mathbb {Z} /p^{i}\mathbb {Z} } 891:-adic numbers fundamental for number theory are completely lacking: they form an 4380: 3623:

Not the order of the digits is reversed, but their production wrt. significance.

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I thought that everybody knows that the standard (in your words the normalized)

325: 4412:-adic representation resembles to positional notation. For example, for every 4401:

As it is, the article contains a method for computing the coefficients of the

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I removed the section "Introduction" with the edit summary "rm the section as

302: 2154:". So I convert this to standard notation. As you possibly see: Your set of 449:

https://www.quantamagazine.org/how-the-towering-p-adic-numbers-work-20201019/

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I'm kind of helpless wrt. your proposed division step consisting of writing

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is often defined by using formal series with a term in each degree of the

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Presently, the article does not contain any understandable definition of

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Remark: I had to correct some process to be repeated indefinitely above.

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color pallete is completely in line with those used in 3d complex plots

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However, if a section would be added on the characters of the group of

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the selected group elements. I could embed, in the image, labels like:

4341:... the simple (but in reversed order) classic binary representation. 903:

is completely lacking, except in a quotation hidden in a footnote; etc.

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The 2-adic integers, with selected corresponding characters on their

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The 3-adic integers, with selected corresponding characters on their

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whose edges are arrows directed toward the root. The nodes of level

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That said, I think the most straightforward improvement would be to

4674:. The subsequent sections still require to be updated and upgraded. 1036:

Now it is about a week that I asked you: What is your «unnormalized

5212: 5138: 4693: 2150:-adic series”. At the beginning you inserted 20:45, 9 June 2021‎ " 2479:{\displaystyle \textstyle \sum _{i=0}^{0}{\tfrac {1}{2}}\,3^{0},} 817:

roots of unity as digits. All such representations are certainly

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general - not just this article - do such a bad job shame on you

4332:{\displaystyle 3=(1,~1\cdot 2,~0\cdot 4,~\ldots )=(1,1,0,...)} 4227:{\displaystyle 2=(0,~1\cdot 2,~0\cdot 4,~\ldots )=(0,1,0,...)} 4122:{\displaystyle 1=(1,~0\cdot 2,~0\cdot 4,~\ldots )=(1,0,0,...)} 2231:. This is not the case in WP, where the more technical use of 701:

Knuth e.g. likes very much the coefficients from the interval

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The order of the sections was confusing. In particular, the

2401:-formal (≈ analytic ??) limits, e.g. when expanding 1/2 for 1564:

are normalized 5-adic series. Their difference as series is

1557:{\displaystyle \textstyle \sum _{i=0}^{\infty }3\cdot 5^{i}} 2329:-adic series" is exactly the same concept as the "infinite 1058:) ― so what is the gain in introducing these unnormalized 452:

https://divisbyzero.com/2008/11/24/what-are-p-adic-numbers/

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I saw your repair wrt. numerator. Indeed, great insight !!

1081:-adic series are unavoidables as soon as one compute with 1054:-adic series» (as I stimulated you in this talk's section 3292:{\displaystyle {\frac {1}{2}}=2+3\cdot 1-{\frac {9}{2}},} 2177:

It is wrong to say that using formal series for defining

1009:!! And they do that absolutely without your unnormalized 1901:{\displaystyle a_{i}\in \mathbb {Z} /p^{i}\mathbb {Z} ,} 4920:(in decimal notation or in base 3-notation or both) to 4708: 2486:

a valid 3-adic expansion of the rational number 1/2 ??

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that might inspire a slightly more elementary image. –

4626:-adic series was described before the definition of a 3730:

Show the order of the digits in the standard notation.

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By the way, I do not like so much the introduction of

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Remove misplaced "with the order of digits reversed".

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only if this was presented as a standard definition.

324:, a collaborative effort to improve the coverage of 2185:, as many textbooks use formal series for defining 1226:{\displaystyle \textstyle p=3,a={\frac {1}{2}},b=1} 868:-adic series are needed for defining operations on 5095: 5024: 4991: 4965: 4945: 4912: 4558: 4371: 4331: 4226: 4121: 4014: 3971: 3938: 3776: 3719: 3542: 3515: 3488: 3426: 3399: 3291: 3230: 3179: 3153: 3116: 3064: 3009: 2966: 2921: 2884: 2827: 2784: 2741: 2685: 2562: 2532: 2478: 2389: 2036: 1960: 1927: 1900: 1838: 1744: 1556: 1500: 1433: 1266: 1225: 1144: 807: 766: 737: 693: 640: 613: 551: 5096:{\displaystyle 25=221_{3}\to 7=21_{3}\to 1=1_{3}} 1043:"In this talk", let me call it the „unnormalized 4892:For a comprehensive image, I would use a rooted 2189:-adic numbers. The definition/representation of 33:for general discussion of the article's subject. 5319:Knowledge level-5 vital articles in Mathematics 2260:You saw my remark and reverted it. But this is 4744:(edit: this was wrong in the original comment) 3489:{\displaystyle a_{i}={\tfrac {n_{i}}{d_{i}}},} 2579:As far as I can see: especially your section « 614:{\displaystyle a_{i}={\tfrac {n_{i}}{d_{i}}},} 5268:So... what kind of colors would you suggest? 5217:Proof that I don't always use the same colors 3019:(you forgot this condition in your post). As 2415:-adic series.» Why that? Isn't the (I admit: 1267:{\displaystyle \textstyle a+b={\frac {3}{2}}} 425:This page has archives. Sections older than 174: 8: 4630:-adic series. So, I have written a section " 4622:-adic expansion of a rational number into a 4408:Also, it is only for positive integers that 962:You still do not give a reference for your « 528:The new section defines the coefficients by 3299:and the process can be repeated infinitely. 493:is prime. So, introducing the subject with 4999:should correspond to the reduction modulo 3400:{\displaystyle \sum _{i=k}^{k}a_{i}p^{i},} 2211:for including the general definition of a 270: 5087: 5068: 5049: 5037: 5010: 5004: 4978: 4958: 4931: 4925: 4905: 4541: 4504: 4429: 4363: 4357: 4240: 4135: 4030: 4006: 4000: 3963: 3957: 3918: 3905: 3880: 3867: 3848: 3835: 3829: 3768: 3762: 3711: 3689: 3680: 3639: 3634: 3534: 3528: 3507: 3501: 3474: 3464: 3457: 3448: 3442: 3418: 3412: 3388: 3378: 3368: 3357: 3351: 3276: 3245: 3243: 3220: 3200: 3192: 3166: 3143: 3129: 3106: 3086: 3081: 3024: 2986: 2980: 2940: 2934: 2908: 2897: 2871: 2812: 2810: 2761: 2755: 2722: 2671: 2654: 2554: 2548: 2518: 2512: 2466: 2448: 2442: 2431: 2424: 2375: 2369: 2030: 2029: 2023: 2014: 2010: 2009: 2002: 2001: 1989: 1980: 1976: 1975: 1973: 1946: 1940: 1919: 1913: 1891: 1890: 1884: 1875: 1871: 1870: 1861: 1855: 1821: 1808: 1802: 1733: 1711: 1700: 1687: 1662: 1651: 1638: 1622: 1611: 1598: 1588: 1577: 1571: 1547: 1531: 1520: 1513: 1491: 1481: 1470: 1463: 1425: 1412: 1399: 1386: 1375: 1362: 1352: 1342: 1331: 1318: 1308: 1298: 1287: 1281: 1253: 1238: 1200: 1179: 1136: 1132: 1131: 1128: 872:-adic numbers, specially for division of 788: 750: 706: 665: 632: 626: 599: 589: 582: 573: 567: 543: 537: 2135:-adic series” is WP:OR=Original Research 1778:-adic series. But the representation of 955:-adic numbers, even not for division of 932:Very good that you do not intend to use 5309:Knowledge vital articles in Mathematics 4900:should be labeled by the integers from 4350:article's definition of p-adic_integers 2665: 2460: 936:-adic series for the representation of 853:-adic series for the representation of 435:when more than 10 sections are present. 272: 231: 5188: 4821: 3065:{\displaystyle 1=2\cdot 2+(-1)\cdot 3} 1165:being closed under these operations ?? 5324:B-Class vital articles in Mathematics 5159:-adic numbers should put some set of 3340:Sorry, I still can't resist: Isn't a 7: 3991:The reader can understand that, for 3594:-adic expansion of rational numbers” 3562:-adic series», and a simpler one ?? 2321:About the terminology, I know that " 1839:{\displaystyle a_{1},a_{2},\ldots ,} 1085:-adic numbers, since the normalized 318:This article is within the scope of 2503:-adic series is the lowest integer 1162:-adic series of D.Lazard type), and 261:It is of interest to the following 23:for discussing improvements to the 2605:. Also, I have clarified that the 2086:-adic numbers are identified with 1712: 1663: 1623: 1589: 1532: 1482: 951:needed for defining operations on 14: 5334:Mid-priority mathematics articles 3979:are integers from {0, 1, . . . , 3821:, that is not used but is valid: 3792:as the first natural numbers... 3626:Make explicit the "integer part". 857:-adic numbers. I'll use only the 429:may be automatically archived by 338:Knowledge:WikiProject Mathematics 5304:Knowledge level-5 vital articles 4665: 3806:Alternative definition is valid? 2828:{\displaystyle {\overline {1}}2} 1145:{\displaystyle \mathbb {Q} _{p}} 1047:-adic series of D.Lazard type“. 386: 341:Template:WikiProject Mathematics 305: 295: 274: 241: 232: 201: 45:Click here to start a new topic. 4953:The arrow from a node of level 2616:An example of computation of a 2082:In many introducing textbooks, 2044:(this is the definition of the 358:This article has been rated as 5314:B-Class level-5 vital articles 5074: 5055: 4531: 4519: 4494: 4482: 4470: 4458: 4452: 4440: 4326: 4296: 4290: 4248: 4221: 4191: 4185: 4143: 4116: 4086: 4080: 4038: 3590:Trying to rescue the section „ 3053: 3044: 2998: 2992: 2952: 2946: 2773: 2767: 2006: 1726: 1717: 1680: 1668: 1418: 1392: 1017:-adic series» be good for ?? ― 864:-adic series. However general 802: 790: 732: 708: 685: 667: 1: 4645:-adic series is not always a 4641:-adic series, the sum of two 3755:P-adic_number#p-adic_integers 3610:to its role in the iteration. 1093:-adic series provide general 332:and see a list of open tasks. 42:Put new text under old text. 5329:B-Class mathematics articles 5193:something like my ideas here 3802:of valid values as example. 3010:{\displaystyle v_{3}(r): --> 2817: 2785:{\displaystyle v_{3}(s): --> 2749:, but appear not to satisfy 2742:{\displaystyle 0\leq a<3} 2686:{\displaystyle r=a\,p^{k}+s} 2533:{\displaystyle a_{i}\neq 0.} 2390:{\displaystyle d_{i}\not =1} 5278:04:59, 4 October 2023 (UTC) 5261:02:22, 4 October 2023 (UTC) 5248:19:03, 3 October 2023 (UTC) 5204:05:11, 3 October 2023 (UTC) 5184:00:29, 3 October 2023 (UTC) 5135:22:42, 2 October 2023 (UTC) 5113:20:07, 2 October 2023 (UTC) 4888:17:36, 2 October 2023 (UTC) 4876:There are some examples at 4872:17:25, 2 October 2023 (UTC) 4859:09:34, 2 October 2023 (UTC) 4812:01:27, 2 October 2023 (UTC) 4798:00:34, 2 October 2023 (UTC) 4684:15:00, 19 August 2023 (UTC) 4659:16:48, 16 August 2023 (UTC) 4398:-adic numbers and integers. 3231:{\displaystyle -1/2=1-3/2.} 2967:{\displaystyle v_{3}(r)=0,} 2587:and even a defective one. ― 823:Other books start with the 774:an approach which he calls 50:New to Knowledge? Welcome! 5350: 3757:is not evident how to use 3346:formal series of the form 3187:At the next step, one has 3117:{\displaystyle 1/2=2-3/2.} 2419:normalized) 3-adic series 2207:By the way, I have edited 780:Teichmüller representation 621:such that the denominator 4606:20:21, 26 July 2023 (UTC) 4593:18:02, 24 July 2023 (UTC) 4575:13:31, 24 July 2023 (UTC) 4389:14:38, 23 July 2023 (UTC) 3744:18:33, 23 June 2021 (UTC) 3574:14:42, 11 June 2021 (UTC) 3336:14:16, 11 June 2021 (UTC) 2862:18:33, 10 June 2021 (UTC) 2701:=3. What do you propose: 2630:14:39, 10 June 2021 (UTC) 2597:08:42, 10 June 2021 (UTC) 2274:13:23, 20 June 2021 (UTC) 2245:11:26, 20 June 2021 (UTC) 2172:09:32, 20 June 2021 (UTC) 2119:18:34, 13 June 2021 (UTC) 2066:17:18, 13 June 2021 (UTC) 1766:17:18, 13 June 2021 (UTC) 1455:16:07, 13 June 2021 (UTC) 1111:15:26, 13 June 2021 (UTC) 1072:09:43, 13 June 2021 (UTC) 1040:-adic series» good for ? 1007:numerator and denominator 895:of the rationals that is 468:00:25, 9 April 2021 (UTC) 357: 290: 269: 80:Be welcoming to newcomers 4946:{\displaystyle p^{k}-1.} 4420:-adic representation of 4403:normalized p-adic series 4352:, how to obtain a valid 3558:already a kind of your « 2350:21:21, 9 June 2021 (UTC) 2301:16:25, 9 June 2021 (UTC) 2158:-adic series is not the 2054:polynomial factorization 1968:under the canonical map 1027:16:52, 7 June 2021 (UTC) 985:15:43, 7 June 2021 (UTC) 924:08:56, 7 June 2021 (UTC) 844:20:06, 6 June 2021 (UTC) 515:12:15, 30 May 2021 (UTC) 364:project's priority scale 5025:{\displaystyle p^{k-1}} 3749:Plase didactic examples 2974:and thus one must have 1961:{\displaystyle a_{i+1}} 1233:and add, then you have 849:I do not intend to use 321:WikiProject Mathematics 5299:B-Class vital articles 5218: 5148: 5097: 5026: 4993: 4967: 4947: 4914: 4703: 4560: 4373: 4348:Now, returning to the 4333: 4228: 4123: 4016: 3973: 3940: 3778: 3721: 3544: 3517: 3490: 3428: 3401: 3373: 3293: 3232: 3181: 3155: 3154:{\displaystyle s=-3/2} 3118: 3076:for 2 and 3), one has 3066: 3012: 2968: 2923: 2922:{\displaystyle r=1/2,} 2886: 2829: 2787: 2743: 2687: 2564: 2534: 2480: 2447: 2391: 2229:associated graded ring 2038: 1962: 1929: 1902: 1840: 1774:-adic numbers through 1746: 1716: 1667: 1627: 1593: 1558: 1536: 1502: 1486: 1435: 1391: 1347: 1303: 1268: 1227: 1146: 821:-adic series, as well. 809: 768: 739: 695: 642: 615: 553: 474:Section "Introduction" 432:Lowercase sigmabot III 75:avoid personal attacks 5216: 5142: 5098: 5027: 4994: 4968: 4948: 4915: 4697: 4561: 4374: 4372:{\displaystyle x_{i}} 4334: 4229: 4124: 4017: 4015:{\displaystyle a_{i}} 3974: 3972:{\displaystyle a_{i}} 3941: 3779: 3777:{\displaystyle x_{i}} 3722: 3545: 3543:{\displaystyle d_{i}} 3518: 3516:{\displaystyle n_{i}} 3491: 3429: 3427:{\displaystyle a_{i}} 3407:where every nonzero 3402: 3353: 3294: 3233: 3182: 3156: 3119: 3067: 3013: 2969: 2924: 2887: 2830: 2788: 2744: 2717:=1/2 ?? Both satisfy 2688: 2565: 2563:{\displaystyle n_{i}} 2535: 2481: 2427: 2392: 2039: 1963: 1930: 1928:{\displaystyle a_{i}} 1903: 1846:such that, for every 1841: 1747: 1696: 1647: 1607: 1573: 1559: 1516: 1503: 1466: 1436: 1371: 1327: 1283: 1269: 1228: 1147: 810: 808:{\displaystyle (p-1)} 769: 740: 696: 643: 641:{\displaystyle d_{i}} 616: 554: 552:{\displaystyle a_{i}} 248:level-5 vital article 195:Auto-archiving period 100:Neutral point of view 5036: 5003: 4977: 4957: 4924: 4904: 4428: 4356: 4239: 4134: 4029: 3999: 3983:− 1}. So, seems the 3956: 3828: 3761: 3633: 3527: 3500: 3441: 3411: 3350: 3242: 3191: 3180:{\displaystyle a=2.} 3165: 3128: 3080: 3023: 2979: 2933: 2896: 2870: 2809: 2754: 2721: 2653: 2547: 2511: 2493:-adic valuation, or 2423: 2368: 2235:has been preferred. 2048:-adic numbers as an 1972: 1939: 1912: 1854: 1801: 1570: 1512: 1462: 1280: 1237: 1178: 1127: 1001:-adic operations of 943:But certainly, your 787: 767:{\displaystyle p=3,} 749: 705: 664: 648:is not divisible by 625: 566: 536: 344:mathematics articles 105:No original research 4992:{\displaystyle k-1} 4973:to a node of level 4878:this Quanta article 3985:positional notation 3952:is a prime and the 2885:{\displaystyle p=3} 2209:Formal power series 2160:Formal power series 1794:-adic integer as a 5219: 5149: 5093: 5022: 4989: 4963: 4943: 4910: 4704: 4556: 4369: 4329: 4224: 4119: 4012: 3969: 3936: 3774: 3727:has to be omitted. 3717: 3616:Make the value of 3540: 3513: 3496:such that none of 3486: 3481: 3424: 3397: 3289: 3228: 3177: 3151: 3114: 3062: 3007: 2964: 2919: 2882: 2825: 2782: 2739: 2683: 2666: 2560: 2530: 2476: 2475: 2461: 2458: 2407:You say: «So, the 2387: 2196:The definition of 2034: 1958: 1925: 1898: 1836: 1742: 1554: 1553: 1498: 1497: 1431: 1264: 1263: 1223: 1222: 1142: 901:discrete valuation 805: 764: 735: 691: 638: 611: 606: 549: 313:Mathematics portal 257:content assessment 86:dispute resolution 47: 4966:{\displaystyle k} 4913:{\displaystyle 0} 4286: 4272: 4258: 4181: 4167: 4153: 4076: 4062: 4048: 3811:This Gupta's link 3702: 3647: 3480: 3284: 3253: 3074:Bézout's identity 2820: 2583:-adic series» is 2457: 2310:-adic series are 2181:-adic numbers is 1782:-adic numbers as 1261: 1208: 694:{\displaystyle .} 605: 439: 438: 378: 377: 374: 373: 370: 369: 226: 225: 66:Assume good faith 43: 5341: 5102: 5100: 5099: 5094: 5092: 5091: 5073: 5072: 5054: 5053: 5031: 5029: 5028: 5023: 5021: 5020: 4998: 4996: 4995: 4990: 4972: 4970: 4969: 4964: 4952: 4950: 4949: 4944: 4936: 4935: 4919: 4917: 4916: 4911: 4826:to understanding 4824:illustrative aid 4787: 4710: 4673: 4669: 4668: 4581:two's complement 4565: 4563: 4562: 4557: 4546: 4545: 4509: 4508: 4423: 4397: 4378: 4376: 4375: 4370: 4368: 4367: 4338: 4336: 4335: 4330: 4285: 4271: 4257: 4233: 4231: 4230: 4225: 4180: 4166: 4152: 4128: 4126: 4125: 4120: 4075: 4061: 4047: 4021: 4019: 4018: 4013: 4011: 4010: 3978: 3976: 3975: 3970: 3968: 3967: 3945: 3943: 3942: 3937: 3923: 3922: 3910: 3909: 3885: 3884: 3872: 3871: 3853: 3852: 3840: 3839: 3817:definition for 3813:has a, perhaps, 3783: 3781: 3780: 3775: 3773: 3772: 3753:For example for 3726: 3724: 3723: 3718: 3716: 3715: 3703: 3698: 3690: 3685: 3684: 3648: 3640: 3619: 3609: 3602: 3553: 3550:is divisible by 3549: 3547: 3546: 3541: 3539: 3538: 3522: 3520: 3519: 3514: 3512: 3511: 3495: 3493: 3492: 3487: 3482: 3479: 3478: 3469: 3468: 3459: 3453: 3452: 3433: 3431: 3430: 3425: 3423: 3422: 3406: 3404: 3403: 3398: 3393: 3392: 3383: 3382: 3372: 3367: 3298: 3296: 3295: 3290: 3285: 3277: 3254: 3246: 3237: 3235: 3234: 3229: 3224: 3204: 3186: 3184: 3183: 3178: 3160: 3158: 3157: 3152: 3147: 3123: 3121: 3120: 3115: 3110: 3090: 3071: 3069: 3068: 3063: 3018: 3015: 3014: 3008: 2991: 2990: 2973: 2971: 2970: 2965: 2945: 2944: 2928: 2926: 2925: 2920: 2912: 2891: 2889: 2888: 2883: 2834: 2832: 2831: 2826: 2821: 2813: 2801: 2797: 2793: 2790: 2789: 2783: 2766: 2765: 2748: 2746: 2745: 2740: 2716: 2712: 2708: 2704: 2700: 2696: 2692: 2690: 2689: 2684: 2676: 2675: 2619: 2608: 2604: 2573: 2570:is divisible by 2569: 2567: 2566: 2561: 2559: 2558: 2539: 2537: 2536: 2531: 2523: 2522: 2506: 2502: 2496: 2492: 2485: 2483: 2482: 2477: 2471: 2470: 2459: 2450: 2446: 2441: 2414: 2410: 2404: 2396: 2394: 2393: 2388: 2380: 2379: 2259: 2145: 2102: 2098: 2089: 2085: 2047: 2043: 2041: 2040: 2035: 2033: 2028: 2027: 2018: 2013: 2005: 2000: 1999: 1984: 1979: 1967: 1965: 1964: 1959: 1957: 1956: 1935:is the image of 1934: 1932: 1931: 1926: 1924: 1923: 1907: 1905: 1904: 1899: 1894: 1889: 1888: 1879: 1874: 1866: 1865: 1849: 1845: 1843: 1842: 1837: 1826: 1825: 1813: 1812: 1793: 1789: 1785: 1781: 1777: 1773: 1751: 1749: 1748: 1743: 1738: 1737: 1715: 1710: 1692: 1691: 1666: 1661: 1643: 1642: 1626: 1621: 1603: 1602: 1592: 1587: 1563: 1561: 1560: 1555: 1552: 1551: 1535: 1530: 1507: 1505: 1504: 1499: 1496: 1495: 1485: 1480: 1440: 1438: 1437: 1432: 1430: 1429: 1417: 1416: 1404: 1403: 1390: 1385: 1367: 1366: 1357: 1356: 1346: 1341: 1323: 1322: 1313: 1312: 1302: 1297: 1273: 1271: 1270: 1265: 1262: 1254: 1232: 1230: 1229: 1224: 1209: 1201: 1173: 1161: 1151: 1149: 1148: 1143: 1141: 1140: 1135: 1122: 1100: 1096: 1092: 1088: 1084: 1080: 1061: 1053: 1046: 1039: 1016: 1012: 1003:rational numbers 1000: 947:-adic series is 816: 814: 812: 811: 806: 776:balanced ternary 773: 771: 770: 765: 744: 742: 741: 738:{\displaystyle } 736: 700: 698: 697: 692: 651: 647: 645: 644: 639: 637: 636: 620: 618: 617: 612: 607: 604: 603: 594: 593: 584: 578: 577: 558: 556: 555: 550: 548: 547: 499: 492: 434: 418: 390: 382: 346: 345: 342: 339: 336: 315: 310: 309: 299: 292: 291: 286: 278: 271: 254: 245: 244: 237: 236: 228: 220: 206: 205: 196: 179: 178: 164: 95:Article policies 16: 5349: 5348: 5344: 5343: 5342: 5340: 5339: 5338: 5289: 5288: 5145:Pontryagin dual 5083: 5064: 5045: 5034: 5033: 5006: 5001: 5000: 4975: 4974: 4955: 4954: 4927: 4922: 4921: 4902: 4901: 4818:Manual of Style 4781: 4742:−1/8 = …0101_3 4707: 4700:Pontryagin dual 4692: 4666: 4664: 4616: 4537: 4500: 4426: 4425: 4421: 4395: 4359: 4354: 4353: 4237: 4236: 4132: 4131: 4027: 4026: 4002: 3997: 3996: 3959: 3954: 3953: 3914: 3901: 3876: 3863: 3844: 3831: 3826: 3825: 3819:p-adic integers 3808: 3796: 3764: 3759: 3758: 3751: 3707: 3691: 3676: 3631: 3630: 3617: 3607: 3603:to the formula. 3600: 3596: 3551: 3530: 3525: 3524: 3503: 3498: 3497: 3470: 3460: 3444: 3439: 3438: 3436:rational number 3414: 3409: 3408: 3384: 3374: 3348: 3347: 3326:Best regards. ― 3240: 3239: 3189: 3188: 3163: 3162: 3126: 3125: 3078: 3077: 3021: 3020: 2982: 2976: 2975: 2936: 2931: 2930: 2894: 2893: 2868: 2867: 2807: 2806: 2799: 2795: 2757: 2751: 2750: 2719: 2718: 2714: 2710: 2706: 2702: 2698: 2694: 2667: 2651: 2650: 2617: 2606: 2602: 2571: 2550: 2545: 2544: 2514: 2509: 2508: 2504: 2500: 2494: 2490: 2462: 2421: 2420: 2412: 2408: 2402: 2371: 2366: 2365: 2284: 2282:Valuation first 2253: 2139: 2137: 2100: 2096: 2087: 2083: 2045: 2019: 1985: 1970: 1969: 1942: 1937: 1936: 1915: 1910: 1909: 1880: 1857: 1852: 1851: 1847: 1817: 1804: 1799: 1798: 1791: 1787: 1783: 1779: 1775: 1771: 1729: 1683: 1634: 1594: 1568: 1567: 1543: 1510: 1509: 1487: 1460: 1459: 1421: 1408: 1395: 1358: 1348: 1314: 1304: 1278: 1277: 1235: 1234: 1176: 1175: 1171: 1159: 1130: 1125: 1124: 1120: 1098: 1094: 1090: 1086: 1082: 1078: 1059: 1056:Valuation first 1051: 1044: 1037: 1014: 1010: 998: 914:-adic series". 897:locally compact 893:extension field 785: 784: 783: 778:and which is a 747: 746: 703: 702: 662: 661: 649: 628: 623: 622: 595: 585: 569: 564: 563: 561:rational number 539: 534: 533: 526: 494: 490: 476: 444: 430: 419: 413: 395: 343: 340: 337: 334: 333: 311: 304: 284: 255:on Knowledge's 252: 242: 222: 221: 216: 193: 121: 116: 115: 114: 91: 61: 12: 11: 5: 5347: 5345: 5337: 5336: 5331: 5326: 5321: 5316: 5311: 5306: 5301: 5291: 5290: 5287: 5286: 5285: 5284: 5283: 5282: 5281: 5280: 5266: 5235: 5224: 5220: 5207: 5206: 5186: 5167: 5164: 5153: 5122: 5121: 5120: 5119: 5118: 5117: 5116: 5115: 5090: 5086: 5082: 5079: 5076: 5071: 5067: 5063: 5060: 5057: 5052: 5048: 5044: 5041: 5032:(for example, 5019: 5016: 5013: 5009: 4988: 4985: 4982: 4962: 4942: 4939: 4934: 4930: 4909: 4848: 4844: 4837: 4814: 4778:by 50% or so. 4771: 4770: 4767: 4764: 4761: 4758: 4755: 4754:−1/4 = …0202_3 4752: 4749: 4746: 4740: 4737: 4734: 4731: 4691: 4688: 4687: 4686: 4649:-adic series. 4615: 4612: 4611: 4610: 4609: 4608: 4595: 4585:David Eppstein 4555: 4552: 4549: 4544: 4540: 4536: 4533: 4530: 4527: 4524: 4521: 4518: 4515: 4512: 4507: 4503: 4499: 4496: 4493: 4490: 4487: 4484: 4481: 4478: 4475: 4472: 4469: 4466: 4463: 4460: 4457: 4454: 4451: 4448: 4445: 4442: 4439: 4436: 4433: 4406: 4399: 4366: 4362: 4346: 4345: 4339: 4328: 4325: 4322: 4319: 4316: 4313: 4310: 4307: 4304: 4301: 4298: 4295: 4292: 4289: 4284: 4281: 4278: 4275: 4270: 4267: 4264: 4261: 4256: 4253: 4250: 4247: 4244: 4234: 4223: 4220: 4217: 4214: 4211: 4208: 4205: 4202: 4199: 4196: 4193: 4190: 4187: 4184: 4179: 4176: 4173: 4170: 4165: 4162: 4159: 4156: 4151: 4148: 4145: 4142: 4139: 4129: 4118: 4115: 4112: 4109: 4106: 4103: 4100: 4097: 4094: 4091: 4088: 4085: 4082: 4079: 4074: 4071: 4068: 4065: 4060: 4057: 4054: 4051: 4046: 4043: 4040: 4037: 4034: 4009: 4005: 3995:=2, the valid 3989: 3988: 3966: 3962: 3946: 3935: 3932: 3929: 3926: 3921: 3917: 3913: 3908: 3904: 3900: 3897: 3894: 3891: 3888: 3883: 3879: 3875: 3870: 3866: 3862: 3859: 3856: 3851: 3847: 3843: 3838: 3834: 3807: 3804: 3794: 3771: 3767: 3750: 3747: 3732: 3731: 3728: 3714: 3710: 3706: 3701: 3697: 3694: 3688: 3683: 3679: 3675: 3672: 3669: 3666: 3663: 3660: 3657: 3654: 3651: 3646: 3643: 3638: 3627: 3624: 3621: 3614: 3611: 3604: 3595: 3588: 3587: 3586: 3585: 3584: 3583: 3582: 3581: 3580: 3579: 3578: 3577: 3576: 3563: 3556: 3555: 3554: 3537: 3533: 3510: 3506: 3485: 3477: 3473: 3467: 3463: 3456: 3451: 3447: 3421: 3417: 3396: 3391: 3387: 3381: 3377: 3371: 3366: 3363: 3360: 3356: 3338: 3325: 3323: 3309: 3308: 3307: 3306: 3305: 3304: 3303: 3302: 3301: 3300: 3288: 3283: 3280: 3275: 3272: 3269: 3266: 3263: 3260: 3257: 3252: 3249: 3227: 3223: 3219: 3216: 3213: 3210: 3207: 3203: 3199: 3196: 3176: 3173: 3170: 3150: 3146: 3142: 3139: 3136: 3133: 3113: 3109: 3105: 3102: 3099: 3096: 3093: 3089: 3085: 3061: 3058: 3055: 3052: 3049: 3046: 3043: 3040: 3037: 3034: 3031: 3028: 3006: 3003: 3000: 2997: 2994: 2989: 2985: 2963: 2960: 2957: 2954: 2951: 2948: 2943: 2939: 2918: 2915: 2911: 2907: 2904: 2901: 2881: 2878: 2875: 2850:Quote notation 2842: 2839: 2836: 2824: 2819: 2816: 2781: 2778: 2775: 2772: 2769: 2764: 2760: 2738: 2735: 2732: 2729: 2726: 2682: 2679: 2674: 2670: 2664: 2661: 2658: 2647: 2637: 2636: 2635: 2634: 2633: 2632: 2614: 2577: 2575: 2557: 2553: 2540:» What if the 2529: 2526: 2521: 2517: 2488:You say: «The 2487: 2474: 2469: 2465: 2456: 2453: 2445: 2440: 2437: 2434: 2430: 2406: 2386: 2383: 2378: 2374: 2363: 2353: 2352: 2334: 2319: 2283: 2280: 2279: 2278: 2277: 2276: 2248: 2247: 2215:. This is not 2205: 2194: 2136: 2129: 2128: 2127: 2126: 2125: 2124: 2123: 2122: 2121: 2107: 2104: 2073: 2072: 2071: 2070: 2069: 2068: 2032: 2026: 2022: 2017: 2012: 2008: 2004: 1998: 1995: 1992: 1988: 1983: 1978: 1955: 1952: 1949: 1945: 1922: 1918: 1897: 1893: 1887: 1883: 1878: 1873: 1869: 1864: 1860: 1835: 1832: 1829: 1824: 1820: 1816: 1811: 1807: 1768: 1754: 1753: 1752: 1741: 1736: 1732: 1728: 1725: 1722: 1719: 1714: 1709: 1706: 1703: 1699: 1695: 1690: 1686: 1682: 1679: 1676: 1673: 1670: 1665: 1660: 1657: 1654: 1650: 1646: 1641: 1637: 1633: 1630: 1625: 1620: 1617: 1614: 1610: 1606: 1601: 1597: 1591: 1586: 1583: 1580: 1576: 1550: 1546: 1542: 1539: 1534: 1529: 1526: 1523: 1519: 1494: 1490: 1484: 1479: 1476: 1473: 1469: 1443: 1442: 1441: 1428: 1424: 1420: 1415: 1411: 1407: 1402: 1398: 1394: 1389: 1384: 1381: 1378: 1374: 1370: 1365: 1361: 1355: 1351: 1345: 1340: 1337: 1334: 1330: 1326: 1321: 1317: 1311: 1307: 1301: 1296: 1293: 1290: 1286: 1260: 1257: 1252: 1249: 1246: 1243: 1221: 1218: 1215: 1212: 1207: 1204: 1199: 1196: 1193: 1190: 1187: 1184: 1168: 1167: 1166: 1163: 1139: 1134: 1114: 1113: 1034: 1033: 1032: 1031: 1030: 1029: 990: 989: 988: 987: 968: 967: 966:-adic series». 960: 959:-adic numbers. 941: 940:-adic numbers. 927: 926: 904: 899:; the term of 885:inverse limits 877: 876:-adic numbers. 834:-adic limit. ― 828: 822: 804: 801: 798: 795: 792: 763: 760: 757: 754: 734: 731: 728: 725: 722: 719: 716: 713: 710: 690: 687: 684: 681: 678: 675: 672: 669: 654: 653: 635: 631: 610: 602: 598: 592: 588: 581: 576: 572: 546: 542: 525: 518: 475: 472: 471: 470: 443: 440: 437: 436: 424: 421: 420: 415: 411: 409: 406: 405: 397: 396: 391: 385: 376: 375: 372: 371: 368: 367: 356: 350: 349: 347: 330:the discussion 317: 316: 300: 288: 287: 279: 267: 266: 260: 238: 224: 223: 214: 212: 211: 208: 207: 181: 180: 118: 117: 113: 112: 107: 102: 93: 92: 90: 89: 82: 77: 68: 62: 60: 59: 48: 39: 38: 35: 34: 28: 13: 10: 9: 6: 4: 3: 2: 5346: 5335: 5332: 5330: 5327: 5325: 5322: 5320: 5317: 5315: 5312: 5310: 5307: 5305: 5302: 5300: 5297: 5296: 5294: 5279: 5275: 5271: 5267: 5264: 5263: 5262: 5259: 5256: 5251: 5250: 5249: 5245: 5241: 5236: 5233: 5229: 5225: 5221: 5215: 5211: 5210: 5209: 5208: 5205: 5202: 5199: 5194: 5190: 5187: 5185: 5181: 5177: 5172: 5168: 5165: 5162: 5158: 5154: 5151: 5150: 5146: 5141: 5137: 5136: 5132: 5128: 5114: 5110: 5106: 5088: 5084: 5080: 5077: 5069: 5065: 5061: 5058: 5050: 5046: 5042: 5039: 5017: 5014: 5011: 5007: 4986: 4983: 4980: 4960: 4940: 4937: 4932: 4928: 4907: 4899: 4895: 4891: 4890: 4889: 4886: 4883: 4879: 4875: 4874: 4873: 4870: 4867: 4862: 4861: 4860: 4856: 4852: 4849: 4845: 4842: 4838: 4835: 4831: 4827: 4825: 4819: 4815: 4813: 4810: 4807: 4802: 4801: 4800: 4799: 4795: 4791: 4785: 4779: 4777: 4769:−9 = …22200_3 4768: 4765: 4762: 4760:1/8 = …2122_3 4759: 4757:1/2 = …1112_3 4756: 4753: 4751:1/4 = …2021_3 4750: 4748:−1/2 = …111_3 4747: 4745: 4741: 4738: 4735: 4732: 4729: 4728: 4727: 4725: 4720: 4716: 4714: 4709: 4701: 4696: 4689: 4685: 4681: 4677: 4672: 4663: 4662: 4661: 4660: 4656: 4652: 4648: 4644: 4640: 4635: 4633: 4629: 4625: 4621: 4613: 4607: 4604: 4601: 4596: 4594: 4590: 4586: 4582: 4578: 4577: 4576: 4572: 4568: 4553: 4550: 4547: 4542: 4538: 4534: 4528: 4525: 4522: 4516: 4513: 4510: 4505: 4501: 4497: 4491: 4488: 4485: 4479: 4476: 4473: 4467: 4464: 4461: 4455: 4449: 4446: 4443: 4437: 4434: 4431: 4419: 4415: 4411: 4407: 4404: 4400: 4393: 4392: 4391: 4390: 4386: 4382: 4364: 4360: 4351: 4344: 4340: 4323: 4320: 4317: 4314: 4311: 4308: 4305: 4302: 4299: 4293: 4287: 4282: 4279: 4276: 4273: 4268: 4265: 4262: 4259: 4254: 4251: 4245: 4242: 4235: 4218: 4215: 4212: 4209: 4206: 4203: 4200: 4197: 4194: 4188: 4182: 4177: 4174: 4171: 4168: 4163: 4160: 4157: 4154: 4149: 4146: 4140: 4137: 4130: 4113: 4110: 4107: 4104: 4101: 4098: 4095: 4092: 4089: 4083: 4077: 4072: 4069: 4066: 4063: 4058: 4055: 4052: 4049: 4044: 4041: 4035: 4032: 4025: 4024: 4023: 4022:are in {0,1} 4007: 4003: 3994: 3986: 3982: 3964: 3960: 3951: 3947: 3933: 3930: 3927: 3924: 3919: 3915: 3911: 3906: 3902: 3898: 3895: 3892: 3889: 3886: 3881: 3877: 3873: 3868: 3864: 3860: 3857: 3854: 3849: 3845: 3841: 3836: 3832: 3824: 3823: 3822: 3820: 3816: 3812: 3805: 3803: 3801: 3793: 3791: 3787: 3769: 3765: 3756: 3748: 3746: 3745: 3741: 3737: 3729: 3712: 3708: 3704: 3699: 3695: 3692: 3686: 3681: 3677: 3673: 3670: 3667: 3664: 3661: 3658: 3655: 3652: 3649: 3644: 3641: 3636: 3628: 3625: 3622: 3615: 3612: 3605: 3598: 3597: 3593: 3589: 3575: 3571: 3567: 3561: 3557: 3535: 3531: 3508: 3504: 3483: 3475: 3471: 3465: 3461: 3454: 3449: 3445: 3437: 3419: 3415: 3394: 3389: 3385: 3379: 3375: 3369: 3364: 3361: 3358: 3354: 3345: 3344: 3343: 3339: 3337: 3333: 3329: 3321: 3320: 3319: 3318: 3317: 3316: 3315: 3314: 3313: 3312: 3311: 3310: 3286: 3281: 3278: 3273: 3270: 3267: 3264: 3261: 3258: 3255: 3250: 3247: 3225: 3221: 3217: 3214: 3211: 3208: 3205: 3201: 3197: 3194: 3174: 3171: 3168: 3148: 3144: 3140: 3137: 3134: 3131: 3111: 3107: 3103: 3100: 3097: 3094: 3091: 3087: 3083: 3075: 3059: 3056: 3050: 3047: 3041: 3038: 3035: 3032: 3029: 3026: 3004: 3001: 2995: 2987: 2983: 2961: 2958: 2955: 2949: 2941: 2937: 2916: 2913: 2909: 2905: 2902: 2899: 2879: 2876: 2873: 2865: 2864: 2863: 2859: 2855: 2851: 2847: 2843: 2840: 2837: 2822: 2814: 2805: 2779: 2776: 2770: 2762: 2758: 2736: 2733: 2730: 2727: 2724: 2680: 2677: 2672: 2668: 2662: 2659: 2656: 2648: 2645: 2644: 2643: 2642: 2641: 2640: 2639: 2638: 2631: 2627: 2623: 2615: 2612: 2600: 2599: 2598: 2594: 2590: 2586: 2582: 2578: 2555: 2551: 2543: 2527: 2524: 2519: 2515: 2499:of a nonzero 2498: 2472: 2467: 2463: 2454: 2451: 2443: 2438: 2435: 2432: 2428: 2418: 2400: 2384: 2381: 2376: 2372: 2361: 2357: 2356: 2355: 2354: 2351: 2347: 2343: 2339: 2335: 2332: 2328: 2324: 2320: 2317: 2313: 2312:formal series 2309: 2305: 2304: 2303: 2302: 2298: 2294: 2290: 2281: 2275: 2271: 2267: 2263: 2257: 2252: 2251: 2250: 2249: 2246: 2242: 2238: 2234: 2233:inverse limit 2230: 2226: 2222: 2218: 2214: 2210: 2206: 2203: 2199: 2195: 2192: 2188: 2184: 2180: 2176: 2175: 2174: 2173: 2169: 2165: 2161: 2157: 2153: 2152:formal series 2149: 2143: 2134: 2130: 2120: 2116: 2112: 2108: 2105: 2093: 2081: 2080: 2079: 2078: 2077: 2076: 2075: 2074: 2067: 2063: 2059: 2055: 2051: 2050:inverse limit 2024: 2020: 2015: 1996: 1993: 1990: 1986: 1981: 1953: 1950: 1947: 1943: 1920: 1916: 1895: 1885: 1881: 1876: 1867: 1862: 1858: 1833: 1830: 1827: 1822: 1818: 1814: 1809: 1805: 1797: 1769: 1767: 1763: 1759: 1755: 1739: 1734: 1730: 1723: 1720: 1707: 1704: 1701: 1697: 1693: 1688: 1684: 1677: 1674: 1671: 1658: 1655: 1652: 1648: 1644: 1639: 1635: 1631: 1628: 1618: 1615: 1612: 1608: 1604: 1599: 1595: 1584: 1581: 1578: 1574: 1566: 1565: 1548: 1544: 1540: 1537: 1527: 1524: 1521: 1517: 1492: 1488: 1477: 1474: 1471: 1467: 1458: 1457: 1456: 1452: 1448: 1444: 1426: 1422: 1413: 1409: 1405: 1400: 1396: 1387: 1382: 1379: 1376: 1372: 1368: 1363: 1359: 1353: 1349: 1343: 1338: 1335: 1332: 1328: 1324: 1319: 1315: 1309: 1305: 1299: 1294: 1291: 1288: 1284: 1276: 1275: 1258: 1255: 1250: 1247: 1244: 1241: 1219: 1216: 1213: 1210: 1205: 1202: 1197: 1194: 1191: 1188: 1185: 1182: 1169: 1164: 1157: 1156: 1155: 1137: 1118: 1117: 1116: 1115: 1112: 1108: 1104: 1076: 1075: 1074: 1073: 1069: 1065: 1057: 1048: 1041: 1028: 1024: 1020: 1008: 1004: 996: 995: 994: 993: 992: 991: 986: 982: 978: 974: 970: 969: 965: 961: 958: 954: 950: 946: 942: 939: 935: 931: 930: 929: 928: 925: 921: 917: 913: 909: 905: 902: 898: 894: 890: 886: 882: 878: 875: 871: 867: 863: 860: 856: 852: 848: 847: 846: 845: 841: 837: 833: 826: 820: 799: 796: 793: 781: 777: 761: 758: 755: 752: 729: 726: 723: 720: 717: 714: 711: 688: 682: 679: 676: 673: 670: 659: 633: 629: 608: 600: 596: 590: 586: 579: 574: 570: 562: 544: 540: 531: 530: 529: 524:-adic series” 523: 519: 517: 516: 512: 508: 502: 497: 488: 483: 481: 473: 469: 465: 461: 456: 455: 454: 453: 450: 441: 433: 428: 423: 422: 408: 407: 404: 403: 399: 398: 394: 389: 384: 383: 380: 365: 361: 355: 352: 351: 348: 331: 327: 323: 322: 314: 308: 303: 301: 298: 294: 293: 289: 283: 280: 277: 273: 268: 264: 258: 250: 249: 239: 235: 230: 229: 210: 209: 204: 200: 192: 189: 187: 183: 182: 177: 173: 170: 167: 163: 159: 155: 152: 149: 146: 143: 140: 137: 134: 131: 127: 124: 123:Find sources: 120: 119: 111: 110:Verifiability 108: 106: 103: 101: 98: 97: 96: 87: 83: 81: 78: 76: 72: 69: 67: 64: 63: 57: 53: 52:Learn to edit 49: 46: 41: 40: 37: 36: 32: 26: 25:P-adic number 22: 18: 17: 5231: 5227: 5170: 5160: 5156: 5123: 4897: 4894:ternary tree 4840: 4833: 4829: 4780: 4776:WP:THUMBSIZE 4772: 4766:−3 = …2220_3 4743: 4723: 4721: 4717: 4712: 4705: 4670: 4646: 4642: 4638: 4636: 4631: 4627: 4623: 4619: 4617: 4417: 4413: 4409: 4402: 4347: 4342: 3992: 3990: 3980: 3949: 3818: 3814: 3809: 3799: 3797: 3789: 3785: 3752: 3733: 3629:The sign in 3591: 3559: 3341: 2845: 2803: 2610: 2580: 2541: 2416: 2398: 2359: 2337: 2330: 2326: 2322: 2315: 2307: 2288: 2285: 2261: 2213:power series 2197: 2190: 2186: 2178: 2155: 2147: 2138: 2132: 2091: 1055: 1049: 1042: 1035: 972: 963: 956: 952: 948: 944: 937: 933: 911: 907: 888: 880: 873: 869: 865: 861: 858: 854: 850: 831: 824: 818: 657: 655: 527: 521: 503: 495: 486: 484: 477: 458:signature.-- 445: 426: 400: 392: 379: 360:Mid-priority 359: 319: 285:Mid‑priority 263:WikiProjects 246: 198: 184: 171: 165: 157: 150: 144: 138: 132: 122: 94: 19:This is the 4847:expansions. 4763:−1 = …222_3 3800:show a list 2497:-adic order 442:lousy intro 335:Mathematics 326:mathematics 282:Mathematics 148:free images 31:not a forum 5293:Categories 4816:Knowledge 4690:Lead image 3784:. Suppose 3736:Nomen4Omen 3566:Nomen4Omen 3328:Nomen4Omen 3017:0}" /: --> 2854:Nomen4Omen 2792:0}" /: --> 2589:Nomen4Omen 2507:such that 2293:Nomen4Omen 2266:Nomen4Omen 2262:NOT AT ALL 2225:local ring 2221:completion 2164:Nomen4Omen 2111:Nomen4Omen 1447:Nomen4Omen 1064:Nomen4Omen 1019:Nomen4Omen 977:Nomen4Omen 859:normalized 836:Nomen4Omen 782:using the 5255:jacobolus 5198:jacobolus 4882:jacobolus 4866:jacobolus 4806:jacobolus 4733:9 = 100_3 4600:jacobolus 3620:explicit. 2709:=−1/2 or 2697:=1/2 and 2542:numerator 2219:, as the 2131:Section „ 520:Section „ 487:introduce 251:is rated 88:if needed 71:Be polite 21:talk page 5270:Melchoir 5240:Melchoir 5176:Melchoir 5105:D.Lazard 4851:D.Lazard 4790:Melchoir 4784:D.Lazard 4736:3 = 10_3 4676:D.Lazard 4651:D.Lazard 4567:D.Lazard 3815:didactic 2978:0}": --> 2929:one has 2753:0}": --> 2622:D.Lazard 2342:D.Lazard 2256:D.Lazard 2237:D.Lazard 2142:D.Lazard 2058:D.Lazard 1796:sequence 1758:D.Lazard 1103:D.Lazard 1077:General 916:D.Lazard 507:D.Lazard 427:365 days 393:Archives 199:365 days 186:Archives 56:get help 29:This is 27:article. 5171:already 4739:1 = 1_3 4730:0 = 0_3 3798:Please 3788:=2 and 2798:=2 and 2713:=0 and 2705:=1 and 2092:no need 532:«every 362:on the 253:B-class 154:WP refs 142:scholar 4416:, the 4381:Krauss 4343:Is it? 3948:where 3342:finite 2360:formal 259:scale. 126:Google 5147:group 4820:says 4724:label 4702:group 4614:OOOPS 3606:Bind 3599:Bind 3434:is a 3002:: --> 2846:valid 2794:. Or 2777:: --> 2613:− 1). 2585:WP:OR 2336:Also 2223:of a 2217:WP:OR 2202:WP:OR 2183:WP:OR 1154:field 559:is a 480:WP:OR 240:This 169:JSTOR 130:books 84:Seek 5274:talk 5244:talk 5232:want 5180:talk 5131:talk 5127:Qsdd 5109:talk 4855:talk 4794:talk 4680:talk 4671:Done 4655:talk 4589:talk 4571:talk 4385:talk 3740:talk 3570:talk 3523:and 3332:talk 3238:So, 3161:and 2892:and 2858:talk 2734:< 2693:for 2626:talk 2593:talk 2405:=3. 2346:talk 2297:talk 2270:talk 2241:talk 2168:talk 2115:talk 2062:talk 1908:and 1762:talk 1508:and 1451:talk 1107:talk 1068:talk 1023:talk 981:talk 920:talk 840:talk 745:for 511:talk 498:= 10 464:talk 162:FENS 136:news 73:and 5258:(t) 5223:it. 5201:(t) 5047:221 4885:(t) 4869:(t) 4809:(t) 4603:(t) 4583:. — 4424:is 3124:So 2866:If 2574:?? 2399:non 2289:all 949:NOT 815:-st 460:agr 354:Mid 176:TWL 5295:: 5276:) 5246:) 5228:is 5182:) 5133:) 5111:) 5075:→ 5066:21 5056:→ 5040:25 5015:− 4984:− 4941:1. 4938:− 4857:) 4796:) 4682:) 4657:) 4591:) 4573:) 4551:… 4535:⋅ 4526:− 4514:… 4498:⋅ 4489:− 4474:⋅ 4465:− 4447:− 4432:− 4422:–1 4387:) 4288:… 4277:⋅ 4263:⋅ 4183:… 4172:⋅ 4158:⋅ 4078:… 4067:⋅ 4053:⋅ 3912:⋅ 3874:⋅ 3855:⋅ 3742:) 3705:⋅ 3693:− 3674:⋅ 3662:⋅ 3637:− 3572:) 3355:∑ 3334:) 3274:− 3268:⋅ 3226:2. 3215:− 3195:− 3175:2. 3138:− 3112:2. 3101:− 3057:⋅ 3048:− 3036:⋅ 3011:0} 2860:) 2818:¯ 2804:is 2786:0} 2728:≤ 2628:) 2595:) 2528:0. 2525:≠ 2429:∑ 2417:un 2348:) 2299:) 2272:) 2243:) 2170:) 2117:) 2064:) 2056:. 2007:→ 1868:∈ 1850:, 1831:… 1764:) 1731:.5 1721:− 1713:∞ 1698:∑ 1675:− 1664:∞ 1649:∑ 1632:⋅ 1624:∞ 1609:∑ 1605:− 1590:∞ 1575:∑ 1541:⋅ 1533:∞ 1518:∑ 1483:∞ 1468:∑ 1453:) 1406:− 1373:∑ 1329:∑ 1325:− 1285:∑ 1152:a 1109:) 1070:) 1025:) 983:) 922:) 842:) 797:− 712:− 680:− 652:». 513:) 466:) 197:: 156:) 54:; 5272:( 5242:( 5196:– 5178:( 5161:p 5157:p 5129:( 5107:( 5089:3 5085:1 5081:= 5078:1 5070:3 5062:= 5059:7 5051:3 5043:= 5018:1 5012:k 5008:p 4987:1 4981:k 4961:k 4933:k 4929:p 4908:0 4898:k 4853:( 4841:p 4834:p 4830:p 4804:– 4792:( 4786:: 4782:@ 4713:p 4678:( 4653:( 4647:p 4643:p 4639:p 4632:p 4628:p 4624:p 4620:p 4598:– 4587:( 4569:( 4554:. 4548:+ 4543:i 4539:p 4532:) 4529:1 4523:p 4520:( 4517:+ 4511:+ 4506:2 4502:p 4495:) 4492:1 4486:p 4483:( 4480:+ 4477:p 4471:) 4468:1 4462:p 4459:( 4456:+ 4453:) 4450:1 4444:p 4441:( 4438:= 4435:1 4418:p 4414:p 4410:p 4396:p 4383:( 4365:i 4361:x 4327:) 4324:. 4321:. 4318:. 4315:, 4312:0 4309:, 4306:1 4303:, 4300:1 4297:( 4294:= 4291:) 4283:, 4280:4 4274:0 4269:, 4266:2 4260:1 4255:, 4252:1 4249:( 4246:= 4243:3 4222:) 4219:. 4216:. 4213:. 4210:, 4207:0 4204:, 4201:1 4198:, 4195:0 4192:( 4189:= 4186:) 4178:, 4175:4 4169:0 4164:, 4161:2 4155:1 4150:, 4147:0 4144:( 4141:= 4138:2 4117:) 4114:. 4111:. 4108:. 4105:, 4102:0 4099:, 4096:0 4093:, 4090:1 4087:( 4084:= 4081:) 4073:, 4070:4 4064:0 4059:, 4056:2 4050:0 4045:, 4042:1 4039:( 4036:= 4033:1 4008:i 4004:a 3993:p 3987:. 3981:p 3965:i 3961:a 3950:p 3934:. 3931:. 3928:. 3925:+ 3920:n 3916:p 3907:n 3903:a 3899:+ 3896:. 3893:. 3890:. 3887:+ 3882:2 3878:p 3869:2 3865:a 3861:+ 3858:p 3850:1 3846:a 3842:+ 3837:0 3833:a 3790:x 3786:p 3770:i 3766:x 3738:( 3734:― 3713:3 3709:5 3700:3 3696:1 3687:+ 3682:2 3678:5 3671:1 3668:+ 3665:5 3659:3 3656:+ 3653:2 3650:= 3645:3 3642:1 3618:s 3608:s 3601:r 3592:p 3568:( 3560:p 3552:p 3536:i 3532:d 3509:i 3505:n 3484:, 3476:i 3472:d 3466:i 3462:n 3455:= 3450:i 3446:a 3420:i 3416:a 3395:, 3390:i 3386:p 3380:i 3376:a 3370:k 3365:k 3362:= 3359:i 3330:( 3287:, 3282:2 3279:9 3271:1 3265:3 3262:+ 3259:2 3256:= 3251:2 3248:1 3222:/ 3218:3 3212:1 3209:= 3206:2 3202:/ 3198:1 3172:= 3169:a 3149:2 3145:/ 3141:3 3135:= 3132:s 3108:/ 3104:3 3098:2 3095:= 3092:2 3088:/ 3084:1 3072:( 3060:3 3054:) 3051:1 3045:( 3042:+ 3039:2 3033:2 3030:= 3027:1 3005:0 2999:) 2996:r 2993:( 2988:3 2984:v 2962:, 2959:0 2956:= 2953:) 2950:r 2947:( 2942:3 2938:v 2917:, 2914:2 2910:/ 2906:1 2903:= 2900:r 2880:3 2877:= 2874:p 2856:( 2823:2 2815:1 2800:s 2796:a 2780:0 2774:) 2771:s 2768:( 2763:3 2759:v 2737:3 2731:a 2725:0 2715:s 2711:a 2707:s 2703:a 2699:p 2695:r 2681:s 2678:+ 2673:k 2669:p 2663:a 2660:= 2657:r 2624:( 2618:p 2611:p 2607:p 2603:p 2591:( 2581:p 2572:p 2556:i 2552:n 2520:i 2516:a 2505:i 2501:p 2495:p 2491:p 2473:, 2468:0 2464:3 2455:2 2452:1 2444:0 2439:0 2436:= 2433:i 2413:p 2409:p 2403:p 2385:1 2382:≠ 2377:i 2373:d 2344:( 2338:p 2331:p 2327:p 2323:p 2316:p 2308:p 2295:( 2268:( 2258:: 2254:@ 2239:( 2198:p 2191:p 2187:p 2179:p 2166:( 2156:p 2148:p 2144:: 2140:@ 2133:p 2113:( 2109:― 2101:p 2097:p 2088:p 2084:p 2060:( 2046:p 2031:Z 2025:i 2021:p 2016:/ 2011:Z 2003:Z 1997:1 1994:+ 1991:i 1987:p 1982:/ 1977:Z 1954:1 1951:+ 1948:i 1944:a 1921:i 1917:a 1896:, 1892:Z 1886:i 1882:p 1877:/ 1872:Z 1863:i 1859:a 1848:i 1834:, 1828:, 1823:2 1819:a 1815:, 1810:1 1806:a 1792:p 1788:p 1784:p 1780:p 1776:p 1772:p 1760:( 1740:. 1735:i 1727:) 1724:2 1718:( 1708:0 1705:= 1702:i 1694:= 1689:i 1685:5 1681:) 1678:3 1672:1 1669:( 1659:0 1656:= 1653:i 1645:= 1640:i 1636:5 1629:3 1619:0 1616:= 1613:i 1600:i 1596:5 1585:0 1582:= 1579:i 1549:i 1545:5 1538:3 1528:0 1525:= 1522:i 1493:i 1489:5 1478:0 1475:= 1472:i 1449:( 1427:i 1423:p 1419:) 1414:i 1410:b 1401:i 1397:a 1393:( 1388:n 1383:k 1380:= 1377:i 1369:= 1364:i 1360:p 1354:i 1350:b 1344:n 1339:k 1336:= 1333:i 1320:i 1316:p 1310:i 1306:a 1300:n 1295:k 1292:= 1289:i 1259:2 1256:3 1251:= 1248:b 1245:+ 1242:a 1220:1 1217:= 1214:b 1211:, 1206:2 1203:1 1198:= 1195:a 1192:, 1189:3 1186:= 1183:p 1172:p 1160:p 1138:p 1133:Q 1121:p 1105:( 1099:p 1095:p 1091:p 1087:p 1083:p 1079:p 1066:( 1060:p 1052:p 1045:p 1038:p 1021:( 1015:p 1011:p 999:p 979:( 973:p 964:p 957:p 953:p 945:p 938:p 934:p 918:( 912:p 908:p 889:p 881:p 874:p 870:p 866:p 862:p 855:p 851:p 838:( 832:p 825:p 819:p 803:) 800:1 794:p 791:( 762:, 759:3 756:= 753:p 733:] 730:1 727:+ 724:, 721:0 718:, 715:1 709:[ 689:. 686:] 683:1 677:p 674:, 671:0 668:[ 658:p 650:p 634:i 630:d 609:, 601:i 597:d 591:i 587:n 580:= 575:i 571:a 545:i 541:a 522:p 509:( 496:p 491:p 462:( 402:1 366:. 265:: 191:1 188:: 172:· 166:· 158:· 151:· 145:· 139:· 133:· 128:( 58:.

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