95:
85:
64:
31:
3422:
decreases. The resulting d is a greatest common divisor, because (at each step) b and r = a – bq have the same divisors as a and b, and thus the same greatest common divisor.....It consists in remarking that the norm N(d) of the greatest common divisor of a and b is a common divisor of N(a), N(b), and N(a + b). When the greatest common divisor D of these three integers has few factors, then it is easy to test, for common divisor, all
Gaussian integers with a norm dividing D.
3465:, this is definitely not true. Not every square of area N contains N grid points, it may be more or less. A counterexample, which is easy to see: A square of the area 3 may contain 1, 2, 3 or 4 gridpoints, depending on its position. I had given a simple geometrical proof for the correct statement, which he has simply deleted. The explanation of the figure, which I provided Knowledge, is also simply removed. No one can understand now, what it means.
3645:: OK, if one uses 'N(a+b)' as third term, my counterexample above does not work. Anyway, your method is not really satisfactory, since it gives only a guess (all Gaussian integers of given norm), an the user has to test all of them anyway, by dividing the given numbers explicitely. Your description of of Euler's algorithm is in my opinion also worse than the previous step-by-step explanation, and I will restore this and the example in the next time.
176:
3608:'s edits have the merit of filling some gaps in the previous version. However, they suffers of several issues. Firstly they limit them to Gauss' original terminology without any link to modern terminology. Also, although Gaussian integers are the basic example for learning algebraic number theory, the distinction was unclear between the properties that was specific to Gaussian integers and the more general properties.
3615:'s edits were an improvement, but were not fully satisfactorily. These are the reasons for which I have rewritten the article. By doing this I have removed some details, because they do not seem really useful. For example, detailed examples for the application of Euclidean algorithm seem not useful, as this duplicates (except for the sub algorithm of Euclidean division) the article
22:
3421:
This algorithm consists of replacing of the input (a, b) by (b, r), where r is the remainder of the
Euclidean division of a by b, and repeating this operation until getting a zero remainder, that is a pair (d, 0). This process terminates, because, at each step, the norm of the second Gaussian integer
2012:
D. Lazard has deleted and rewritten major parts of the article and worsened it with this edits. He has deleted decent computations and examples and replaced them by wrong ones. The text is now much less readable and users will have difficulties to understand the ideas. I had a long going dispute with
1703:
By the way, the previous version of the article was not only incomplete, but also badly structured. The recent edits do not improve the structure, and may be confusing, as the edits emphasize on the Gauss' original point of view, without any connexion with modern knowledge on the subject. The choice
970:
I have only some mathematics training (graduate level electrical engineering), and I was perusing the
Gaussian Moat article out of curiosity and that article had a link pointing here for the definition of Gaussian primes. I see the definition, but do Gaussian primes have the property that you can't
802:
However I still think the original version is badly put. It is a messy mixture of singular and plural. Surely we are not trying to say merely "Every
Gaussian integer is a quadratic integer": we are trying to say something about the Gaussian integers as a structure. So I've had a second go at editing
3763:
has recently added a new section "Congruences and residue classes" to this article. This was lacking. However, this new section was written in a old fashioned style, without any reference to modern terminology. In particular, there was no mention that the congruence classes form the quotient of the
3555:
square, the result is true. I agree that either a proof or a link to the article about lattice would be useful, and I'll add this in a next edit. As the proof is standard in lattice theory, giving detailed proof mixed in a difficult-to-understand geometric description, and not related with lattice
1682:
The citation that has been provided shows that the choice of prime associate is the one that was given by Gauss. However, mathematics have evolved since Gauss, and a citation is still lacking for showing that this choice is standard in modern literature. In fact this choice is somehow problematic,
3619:. On the other hand the use of the norm for improving gcd computation is specific to Gaussian integers (and other rings of algebraic integers), and this deserve to be exemplified (I have given such an example, but others could be useful). Also, examples for Euclidean division could be useful, and
1699:
even). This way of choosing associates is easy to prove and gives the natural associate for real primes. It is thus certainly preferred by modern textbooks, even I have no source at hand for justifying that. Nevertheless, I agree that Gauss's choice should be mentioned, but only has an historical
4156:
OK, the sentence was ambiguous, as "by any element of the ring" should apply only to multiplication. For avoiding the ambiguity, it should be written "an ideal is a subset of a ring that is stable under multiplication by any element of the ring and addition". But this is awkward, and "stable" is
3623:'s example could be useful for that, if adapted for this purpose. This example is not convenient for gcd, as the use of the norm makes it completely trivial. There are certainly more possible improvements of my version, but, as it is, I am convinced that it is better than all preceding versions.
238:." That can't be right. The imaginary part is by definition some multiple of i. Even where b = 0, the imaginary part is a natural number, but still not an integer. Isn't it more correct to say "a complex number where the real part and the argument of the imaginary part are both integers"?
1683:
as, for example, 231 has not the same factorization over the integers and the
Gaussian integers (3.7.11 vs. (–1).(–3).(-7).(–11)). This could be acceptable if there were not a better (an easier) choice for the associate. In fact, every Gaussian integer with an odd norm has a unique associate
2024:
I had written a explanation how to compute the gcd using Euclid algorithm step by step, which was easy to understand for readers and have given an instructive example for that. He deleted this completely and replaced it with useless methods to compute the gcd by using the norm. Here it is:
4000:
3651:: After long dispute you have finally realised, that your claim was wrong, and my proof (which you had deleted) is needed. You have rewritten it, but IMO less explanative and in poorer English. Therefore, I will probably restore the previous version in the next time.
3768:. Also, as it is heavily used that an ideal is a sublattice of the lattice of the Gaussian integer, there was no mention of that. I have thus rewritten this section in a more encyclopedic style (for an encyclopedia, links to related notions are fundamental).
807:, being a particular case of a commutative ring of quadratic integers." I hope I've got it right this time, but if not, perhaps someone could reword it to say something clear about the Gaussian integers as a structure, rather than just reverting.
4001:
https://web.archive.org/web/20120306225505/http://www.imocompendium.com/index.php?options=mbb%7Ctekstkut&page=0&art=extensions_ddj%7Cf&ttn=Dushan%20D%3Bjukic1%7C%20Arithmetic%20in%20Quadratic%20Fields%7CN%2FA&knj=&p=3nbbw45001
862:
used in the caption to the illustration of "Repartition in the plane of the small
Gaussian primes" — which is just a picture of the small Gaussian primes — also giving no hint of what the word means. This is another place the word should not be
1977:
I have severe doubts, that a modern textbook could give a better explanation than Gauß himself, but you may convince me with a reference. I think, these topics are of such fundamental nature, that 'modern research' can hardly give any new
4004:
3950:
3866:
844:
of what the word "repartition" means. Also, I've been around a while and have never seen the word "repartition" used in mathematics or anywhere else. Looking up the word in dictionaries did not help explain its use in this article.
3715:
These assertions are based on the rules of fairness and respect for the work of co-workers, which are surely also anchored somewhere in WP. Your last sentence is right, but you should acknowledge that it applies for you, too.
3301:
3410:
3223:
2533:
1738:
I agree, that there are a plenty of other possibilities to define 'primary', but I disagree, that others are better than Gauß' proposal, even if some might find it 'strange'. I will try to explain the problem.
1332:
385:
3570:
I have not found in
Knowledge articles the basic result of lattice theory which implies that the number of residue classes is the norm of the modulus. Therefore I have provided one essentially derived from
4005:
http://www.imocompendium.com/index.php?options=mbb%7Ctekstkut&page=0&art=extensions_ddj%7Cf&ttn=Dushan%20D%3Bjukic1%7C%20Arithmetic%20in%20Quadratic%20Fields%7CN%2FA&knj=&p=3nbbw45001
3012:
3656:
Your action in the latest edits was not acceptable. You don't have the right to delete decent content and replace it with one, that is wrong or less readable, with the specious justification to make it
2604:
505:. My apologies if I am wrong here as I dont understand this math concept. However, its precisely the reason for my doubt. I hope somebody with proper understanding clarifies this. -- wadkar <AT: -->
3093:
2954:
3776:
Restored the version prior to Lazard's deteriorations, which destroyed the context for the figure. This version is also much more instructive and better readable (discussion see 'talk page')
1390:
151:
2475:
1814:
I am absolutely sure, that Gauß has considered all these ideas (he was a genius, as you probably know). His final definition is to prefer, because it fulfills an important requirement:
4201:
1199:
1130:
4094:: 'A subset I is called a two-sided ideal (or simply an ideal) of R if it is an additive subgroup of R that "absorbs multiplication by elements of R."'. A trivial counterexample is
2703:
2335:
1952:
2888:
2398:
727:
687:
433:
2763:
1981:
Your assertion that the article is badly structured, is wrong. The article is clearly structured and contains most of the important & relevant topics about
Gaussian integers.
1741:
The definition of primary associates which you propose, is surely possible. Yet another simple way is, to choose them from the first quadrant (excluding the imiginary axis), i.e.
1876:
2131:
4117:
4139:
582:
560:
503:
481:
459:
2640:
2269:
2229:
2076:
1077:
2802:
1473:
3689:: On which Knowledge rules are based these assertions? Moreover the fact that your added content was decent and acceptable is just your personal opinion, nothing more.
3334:
3126:
2196:
1430:
979:? That isn't clear to me from this or the prime elements article, and I think spelling it out would make the Gaussian prime section more accessible. - Daniel Morgret
4191:
3596:'s edits, the article was incomplete and badly structured. In particular, although almost every properties result from Euclidean division, this was sketchy described
3039:
2829:
2428:
2158:
1984:
By the way, I am very curious to hear, what lattices have to do with residue classes? If you add an extension to the article, I will surely read it with interest. --
1908:
1768:
1556:
2013:
him on this page (see above), in which he was never able to prove his claims, but despite this he went on with his destruction. Here are two of the worst examples:
1527:
4206:
4032:
4028:
4014:
1972:
1223:
3871:
3787:
3434:, this is unusable for more complicated cases. Simple counterexample: the gcd of 3+2i and 2+3i is not 3+2i (nor 2+3i) what his formula gives (since both have
1808:
855:
Its meaning cannot be inferred from context, either. (Though this would be a very bad way for an encyclopedia article to communicate the meaning of a word.)
848:
I strongly suggest that this sentence be rephrased to avoid the use of that word. Or if for some reason it is important to use that word, then will someone
35:
4216:
141:
3600:
the description the resulting properties. Also, these properties were systematically presented in terms of abstract ring theory, which is unnecessarily
4186:
3228:
483:. Also, the above query/talk/point/discussion mentions that both a, b are integers, while the Latex definition, as per my understanding, puts them in
4090:
Quote: "an ideal is a subset of a ring that is stable under addition and multiplication by any element of the ring". Under addition? No. Check the
4196:
117:
4211:
4142:
3339:
3131:
753:
745:
Someone more knowledgeable than myself should write an article about this ring (or such rings) in case such articles don't already exist.
3468:
There are many other mistakes, and I suggest that these latest edits shoud be undone. Could somebody of the page watchers please help?
986:
621:
2480:
514:
4010:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
108:
69:
3954:
without any explanation of this unusual and undefined terminology and notation cannot be qualified of "more readable of anything".
1228:
315:
748:
A link to this article should be provided within the article about
Gaussian integers, at the very least in the "See also" section.
4181:
2959:
281:, as the argument of any non-zero integer multiple of i is π/2, while its modulus is that integer which you multiplied by i. --
189:
946:
in
English. So both uses are about the distribution of Gaussian primes, which makes a lot more sense. I'll make the change. --
2538:
1704:
of the prime associates is an example. The description of the residue classes without any reference to the modern theory of
1818:. This is the case for his proposal, not for yours (and also not for the 'I. quadrant choice'). A simple counterexample:
1017:, and thus for Gaussian integers. I have added this property to the article, and I hope that this answers your question.
4075:
44:
3044:
1014:
852:
define it in that section, since I expect that at least 99.99% of readers of this article have no idea what it means.
219:
2897:
795:
I tried to change "The Gaussian integers are a special case of the quadratic integers." I got it wrong. Thank you
1337:
2433:
4031:
to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
916:
If you think most users would not understand the word in this context, you could change it to something else. —
198:
4146:
1578:
589:
757:
1148:
990:
625:
4066:
3992:
1082:
518:
2645:
2277:
1913:
4050:
If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
4038:
3604:. Also, the fundamental result, that the norm is the number of residue classes, was completely lacking.
2834:
2340:
951:
812:
775:, in the "See also" section and elsewhere. Perhaps the "See also" links need brief explanatory notes? —
692:
652:
404:
50:
3991:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit
2708:
435:? I thought every complex number can be represented as a tuple of two real numbers. And these Gaussian
94:
298:
Hi, I am confused by the Latex definition given on the page "Formally, Gaussian integers are the set "
1821:
982:
617:
510:
3575:'s one, but, as this is not really specific to Gaussian integers, I have put it in a collapsed box.
2084:
1615:
Please note: this was not my idea (I wish it was), but the idea of Mr. Gauß himself. Please look at
1136:
disputed inline|reason= wrong: a+bi and -a–bi are congruent mod 2+2i and associated|date=August 2017
21:
4091:
3616:
3601:
3533:) = 50, which have the gcd 1, showing that the gcd of 3+2i and 2+3i divides 1 and is therefore one.
2035:
1006:
921:
903:
796:
585:
4097:
1503:
OK, my argument is wrong. But this is a very strange way for saying that every prime different of
116:
on Knowledge. If you would like to participate, please visit the project page, where you can join
4162:
4122:
3966:
3958:
3779:
3778:. I am not willing to discuss the aggressive style of this edit summary (I have already recalled
3739:
3721:
3694:
3682:
3672:
3628:
3580:
3561:
3476:
1989:
1713:
1673:
1638:
1589:
1483:
1022:
835:
Most of the unsolved problems are related to the repartition in the plane of the Gaussian primes.
768:
565:
543:
486:
464:
442:
234:
May I query this: "A Gaussian integer is a complex number whose real and imaginary part are both
204:
100:
4035:
before doing mass systematic removals. This message is updated dynamically through the template
2612:
2234:
2201:
2041:
1619:
of his paper (unfortunately in German), there you will find exactly my statement, given by Gauß:
584:
is standard notation for the set of (real) integers. It is short for the German word "Zahlen". —
84:
63:
4051:
1043:
2768:
1439:
780:
772:
270:
3551:: I have clarified in the article that it is a semi-open square that is considered here. For
1881:
In fact, he gives yet another possibility in his his paper, which fulfills this requirement:
3984:
3782:
to this user, see above in this talk page). I want just to remark that a text that contains
3687:"Your action in the latest edits was not acceptable. You don't have the right to delete ..."
3306:
3098:
2163:
1395:
947:
808:
804:
278:
200:
175:
4058:
767:
That is a good suggestion, and the article already has links to the numbers you mentioned,
3017:
2807:
2406:
2136:
1705:
1010:
3945:{\displaystyle t=\pm {\frac {1}{2}},\pm {\frac {3}{2}},\dots ,\;s=-\infty ,\dots \infty }
3861:{\displaystyle s=\pm {\frac {1}{2}},\pm {\frac {3}{2}},\dots ,\;t=-\infty ,\dots \infty }
3513:(this is the basis of the proof of Euclidean algorithm). Not ignoring the error, we have
1884:
1744:
1532:
1624:
1506:
4017:, "External links modified" talk page sections are no longer generated or monitored by
917:
899:
868:
274:
249:
4057:
If you found an error with any archives or the URLs themselves, you can fix them with
1957:
1208:
649:
I am only an amateur mathematician, and I don't know the term for numbers of the form
4175:
4158:
3962:
3771:
3765:
3760:
3735:
3717:
3690:
3668:
3624:
3620:
3612:
3605:
3593:
3576:
3572:
3557:
3472:
1985:
1709:
1669:
1656:
1634:
1603:
1585:
1498:
1479:
1018:
1002:
741:), but I do know that they are significant. I suggest that these two steps be taken:
253:
239:
1773:
1631:
I think I will include the citation in my next edit, and hope you believe it now...
1571:
1436:(defined in same chapter, see example 1), which are never primes, except the noted
776:
3296:{\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {2}{1+\mathrm {i} }}=1-\mathrm {i} }
971:
factor them? For example, for X = a + bi, is there no set of Gaussian integers Y
4024:
113:
3539:
As the area of this square is N(m), it contains exactly m Gaussian integers....
3455:
As the area of this square is N(m), it contains exactly m Gaussian integers....
4023:. No special action is required regarding these talk page notices, other than
90:
864:
282:
202:
4166:
4150:
4080:
3970:
3743:
3725:
3698:
3676:
3632:
3584:
3565:
3480:
1993:
1717:
1708:
is another example. In a near future, I'll try to remediate these issues.
1677:
1642:
1625:
Carl Friedrich Gauss’ Arithmetische Untersuchungen über höhere Arithmetik.
1593:
1487:
1026:
994:
955:
925:
907:
872:
816:
784:
761:
629:
593:
522:
285:
256:
242:
266:
235:
3734:
above: Your statements are just your personal opinion, nothing more. --
2038:
can be used: For the determination of the gcd of two Gaussian integers
3405:{\displaystyle {\underline {(5+\mathrm {i} ,2)=z_{2}=1+\mathrm {i} }}}
3218:{\displaystyle z_{2}=z_{0}-q_{1}z_{1}=5+\mathrm {i} -4=1+\mathrm {i} }
4157:
unnecessary jargon. Thus I have rewritten things for clarification.
1079:, there is one and only one associate, which fulfils the congruence
2528:{\displaystyle \left|q_{1}-\xi \right|\leq {\frac {1}{\sqrt {2}}}}
1816:
the product of two primary numbers should also be a primary number
1327:{\displaystyle z-(-z)=2z\;{\stackrel {!}{=}}\;q(2+2i)=2q(1+i)}
938:
This is some translation awkwardness by a non-native speaker.
380:{\displaystyle \mathbb {Z} =\{a+bi\mid a,b\in \mathbb {Z} \}.}
205:
169:
15:
1141:
Your statement is wrong. What makes you think, that for an
614:
Thanks, I got confused by the notation and their meaning.
3995:
for additional information. I made the following changes:
3665:
Please stop acting, as if you were the owner of this page.
3007:{\displaystyle {\frac {z_{0}}{z_{1}}}=2.5+0.5\mathrm {i} }
2599:{\displaystyle |z_{2}|\leq {\frac {|z_{1}|}{\sqrt {2}}}}
3988:
1570:
even. As your way for choosing the prime element seems
4125:
4100:
3874:
3790:
3774:
has just restored his version with the edit summary:
3342:
3309:
3231:
3134:
3101:
3047:
3020:
2962:
2900:
2837:
2810:
2771:
2711:
2648:
2615:
2541:
2483:
2436:
2409:
2343:
2280:
2237:
2204:
2166:
2139:
2087:
2044:
1960:
1916:
1887:
1824:
1777:
1747:
1661:
I have included the citation, please remove your tag
1535:
1509:
1442:
1398:
1340:
1231:
1211:
1151:
1085:
1046:
695:
655:
568:
546:
489:
467:
445:
407:
318:
265:
Also, but 0 is an integer, you may be confusing the
112:, a collaborative effort to improve the coverage of
4027:using the archive tool instructions below. Editors
4133:
4111:
3944:
3860:
3541:Ignoring the mistake, that he should have written
3457:Ignoring the mistake, that he should have written
3404:
3328:
3295:
3217:
3120:
3087:
3033:
3006:
2948:
2882:
2823:
2796:
2757:
2697:
2634:
2598:
2527:
2469:
2422:
2392:
2329:
2263:
2223:
2190:
2152:
2125:
2070:
1966:
1946:
1902:
1870:
1801:
1762:
1550:
1521:
1467:
1424:
1384:
1326:
1217:
1193:
1124:
1071:
721:
681:
576:
554:
497:
475:
453:
427:
379:
3088:{\displaystyle 2,2+\mathrm {i} ,3,3+\mathrm {i} }
2430:as a Gaussian integer lying next to the quotient
898:The word has been in use for quite some time. —
4202:Knowledge level-5 vital articles in Mathematics
4013:This message was posted before February 2018.
2949:{\displaystyle z_{0}=5+\mathrm {i} ,\;z_{1}=2}
2477:(there may be up to four). Then always holds
3764:Gaussian integer by an ideal, and no link to
3487:"Ingoring the error, that there should stand
3336:: The algorithm terminates and we get as gcd
2078:it works very similar as for real integers.
1385:{\displaystyle z\;{\stackrel {!}{=}}\;q(1+i)}
803:it, by saying, "The Gaussian integers form a
213:This page has archives. Sections older than
8:
3957:Therefore, I'll revert Wolfk.wk, and notify
3426:Ingoring the error, that there should stand
2470:{\displaystyle \xi :={\frac {z_{0}}{z_{1}}}}
1205:below) says, that there must exist a factor
371:
336:
893:Webster's Seventh New Collegiate Dictionary
3983:I have just modified one external link on
3784:"The grid shall consist of the lines with
3556:theory is misleading and must be avoided.
980:
58:
4127:
4126:
4124:
4105:
4104:
4099:
3900:
3884:
3873:
3816:
3800:
3789:
3391:
3376:
3355:
3343:
3341:
3314:
3308:
3288:
3271:
3259:
3248:
3238:
3232:
3230:
3210:
3190:
3175:
3165:
3152:
3139:
3133:
3106:
3100:
3080:
3060:
3046:
3025:
3019:
2999:
2979:
2969:
2963:
2961:
2934:
2920:
2905:
2899:
2874:
2858:
2845:
2836:
2815:
2809:
2776:
2770:
2750:
2744:
2735:
2727:
2721:
2712:
2710:
2689:
2676:
2666:
2653:
2647:
2620:
2614:
2583:
2577:
2568:
2565:
2557:
2551:
2542:
2540:
2513:
2493:
2482:
2459:
2449:
2443:
2435:
2414:
2408:
2382:
2376:
2367:
2359:
2353:
2344:
2342:
2321:
2308:
2298:
2285:
2279:
2255:
2242:
2236:
2231:there exists a pair of Gaussian integers
2209:
2203:
2165:
2144:
2138:
2117:
2095:
2086:
2062:
2049:
2043:
1959:
1926:
1915:
1886:
1844:
1823:
1776:
1746:
1534:
1508:
1447:
1441:
1414:
1397:
1354:
1349:
1347:
1346:
1339:
1269:
1264:
1262:
1261:
1230:
1210:
1164:
1150:
1095:
1084:
1051:
1045:
705:
694:
665:
654:
570:
569:
567:
548:
547:
545:
491:
490:
488:
469:
468:
466:
447:
446:
444:
421:
420:
406:
367:
366:
320:
319:
317:
3919:
3835:
2928:
1362:
1344:
1277:
1259:
1194:{\displaystyle z\equiv -z{\pmod {2+2i}}}
4192:Knowledge vital articles in Mathematics
1125:{\displaystyle p\equiv 1{\pmod {2+2i}}}
717:
677:
223:when more than 10 sections are present.
60:
19:
4141:, which is not stable under adding 1.
3783:
3775:
3756:Section congruence and residue classes
3686:
3536:
3497:: This is not an error, as the gcd of
3486:
2698:{\displaystyle z_{1}=q_{2}z_{2}+z_{3}}
2330:{\displaystyle z_{0}=q_{1}z_{1}+z_{2}}
4207:C-Class vital articles in Mathematics
7:
3041:therefor the four Gaussian integers
1947:{\displaystyle a\equiv 1{\pmod {4}}}
106:This article is within the scope of
3730:Especially it also applies to your
2883:{\displaystyle (z_{0},z_{1})=z_{n}}
2393:{\displaystyle |z_{2}|<|z_{1}|.}
1608:this is by no means a strange way,
891:a second or additional partition (
722:{\displaystyle a+b{\sqrt {-k}}\ \,}
682:{\displaystyle a+b{\sqrt {-2}}\ \,}
428:{\displaystyle a,b\in \mathbb {I} }
49:It is of interest to the following
4217:High-priority mathematics articles
3939:
3930:
3855:
3846:
3392:
3356:
3289:
3272:
3211:
3191:
3081:
3061:
3000:
2921:
2758:{\displaystyle |z_{3}|<|z_{2}|}
826:The first sentence in the section
14:
3987:. Please take a moment to review
2894:Sought-after shall be the gcd of
1935:
1628:Springer, Berlin 1889, S. 534 ff.
1173:
1143:arbitrary Gaussian integer z=a+ib
1104:
645:Other kinds of "complex integers"
217:may be automatically archived by
126:Knowledge:WikiProject Mathematics
4187:Knowledge level-5 vital articles
2804:. It is easy to see, that then
1871:{\displaystyle (1+2i)^{2}=-3+4i}
174:
129:Template:WikiProject Mathematics
93:
83:
62:
29:
20:
3448:Congruences and residue classes
2126:{\displaystyle (z_{0},0)=z_{0}}
1203:Congruences and residue classes
1040:For all Gaussian primes except
146:This article has been rated as
4197:C-Class level-5 vital articles
3971:13:32, 12 September 2017 (UTC)
3549:, this is definitely not true"
3366:
3346:
2864:
2838:
2751:
2736:
2728:
2713:
2584:
2569:
2558:
2543:
2383:
2368:
2360:
2345:
2179:
2167:
2107:
2088:
1940:
1929:
1841:
1825:
1415:
1411:
1399:
1379:
1367:
1321:
1309:
1297:
1282:
1247:
1238:
1201:? The definition (see chapter
1187:
1167:
1118:
1098:
791:Relation to quadratic integers
540:I don't know what you mean by
330:
324:
1:
4112:{\displaystyle 2\mathbb {Z} }
3744:18:22, 8 September 2017 (UTC)
3726:08:08, 6 September 2017 (UTC)
3699:09:36, 5 September 2017 (UTC)
3677:08:55, 5 September 2017 (UTC)
3633:21:34, 4 September 2017 (UTC)
3585:21:34, 4 September 2017 (UTC)
3566:13:00, 4 September 2017 (UTC)
3481:07:30, 4 September 2017 (UTC)
3095:can be chosen. We chose e.g.
2535:(see above) and consequently
1927:
1432:. That again only holds for
1165:
1096:
1027:11:37, 28 December 2016 (UTC)
995:21:55, 26 December 2016 (UTC)
817:15:12, 15 December 2013 (UTC)
120:and see a list of open tasks.
4212:C-Class mathematics articles
4134:{\displaystyle \mathbb {Z} }
4081:21:07, 11 October 2017 (UTC)
1015:unique factorisation domains
577:{\displaystyle \mathbb {Z} }
555:{\displaystyle \mathbb {I} }
498:{\displaystyle \mathbb {Z} }
476:{\displaystyle \mathbb {I} }
454:{\displaystyle \mathbb {R} }
286:16:37, 5 November 2006 (UTC)
273:. And you may be confusing
2635:{\displaystyle z_{2}\neq 0}
2264:{\displaystyle q_{1},z_{2}}
2224:{\displaystyle z_{1}\neq 0}
2071:{\displaystyle z_{0},z_{1}}
1994:07:34, 21 August 2017 (UTC)
1718:14:05, 20 August 2017 (UTC)
1695:odd and positive (and thus
1678:09:57, 14 August 2017 (UTC)
1643:20:36, 13 August 2017 (UTC)
1594:20:09, 13 August 2017 (UTC)
1488:17:19, 13 August 2017 (UTC)
1013:. The converse is true for
785:14:37, 3 January 2012 (UTC)
762:13:14, 3 January 2012 (UTC)
4233:
4167:11:24, 22 March 2018 (UTC)
4151:10:59, 22 March 2018 (UTC)
4044:(last update: 5 June 2024)
3980:Hello fellow Wikipedians,
1574:, I'll change my tag into
1529:has exactly one associate
1392:This is only the case, if
1072:{\displaystyle p_{1}:=1+i}
956:20:19, 22 April 2016 (UTC)
926:18:50, 22 April 2016 (UTC)
908:18:46, 22 April 2016 (UTC)
873:12:21, 22 April 2016 (UTC)
630:17:07, 27 April 2012 (UTC)
594:22:46, 17 April 2012 (UTC)
523:22:03, 17 April 2012 (UTC)
257:01:55, 13 March 2006 (UTC)
243:01:53, 13 March 2006 (UTC)
2831:is the sought-after gcd:
2797:{\displaystyle z_{n+1}=0}
2642:, this is continued with
2008:Latest edits of D. Lazard
1476:Please remove the dispute
1468:{\displaystyle p_{1}=1+i}
461:or to be precise the set
145:
78:
57:
2019:Greatest common divisor:
1001:I is always true that a
152:project's priority scale
3976:External links modified
3329:{\displaystyle z_{3}=0}
3303:, i.e., the residue is
3225:. The next step gives
3121:{\displaystyle q_{1}=2}
2403:To get this, one takes
2191:{\displaystyle (0,0)=0}
1610:it is original research
1425:{\displaystyle (1+i)|z}
109:WikiProject Mathematics
4182:C-Class vital articles
4135:
4113:
3961:for some arbitration.
3946:
3862:
3685:, and think about it.
3436:N(3+2i) = N(2+3i) = 13
3406:
3330:
3297:
3219:
3122:
3089:
3035:
3008:
2950:
2884:
2825:
2798:
2759:
2699:
2636:
2600:
2529:
2471:
2424:
2394:
2331:
2265:
2225:
2192:
2154:
2127:
2072:
1968:
1948:
1904:
1872:
1804:
1764:
1552:
1523:
1469:
1434:even Gaussian integers
1426:
1386:
1328:
1219:
1195:
1126:
1073:
723:
683:
578:
556:
499:
477:
455:
429:
381:
252:of 2+3i is 3, not 3i.
220:Lowercase sigmabot III
4136:
4114:
3947:
3863:
3416:Lazard's replacement:
3407:
3331:
3298:
3220:
3123:
3090:
3036:
3034:{\displaystyle q_{1}}
3009:
2951:
2885:
2826:
2824:{\displaystyle z_{n}}
2799:
2765:a.s.o, until finally
2760:
2700:
2637:
2601:
2530:
2472:
2425:
2423:{\displaystyle q_{1}}
2395:
2332:
2266:
2226:
2193:
2155:
2153:{\displaystyle z_{0}}
2128:
2073:
1969:
1949:
1905:
1873:
1805:
1802:{\displaystyle a: -->
1765:
1564:odd and positive and
1553:
1524:
1470:
1427:
1387:
1329:
1220:
1196:
1127:
1074:
724:
684:
579:
557:
500:
478:
456:
430:
382:
36:level-5 vital article
4123:
4098:
4092:article about ideals
4025:regular verification
3872:
3788:
3649:Number of gridpoints
3340:
3307:
3229:
3132:
3099:
3045:
3018:
2960:
2898:
2835:
2808:
2769:
2709:
2646:
2613:
2539:
2481:
2434:
2407:
2341:
2278:
2235:
2202:
2164:
2137:
2085:
2042:
1958:
1914:
1903:{\displaystyle a+bi}
1885:
1822:
1775:
1763:{\displaystyle a+bi}
1745:
1551:{\displaystyle a+ib}
1533:
1507:
1440:
1396:
1338:
1229:
1209:
1149:
1083:
1044:
693:
653:
566:
544:
487:
465:
443:
405:
316:
132:mathematics articles
4015:After February 2018
3617:Euclidean algorithm
2036:Euclidean algorithm
1522:{\displaystyle 1+i}
1007:irreducible element
858:Note: This word is
769:Eisenstein integers
689:(or more generally
439:ought to belong to
4131:
4109:
4069:InternetArchiveBot
4020:InternetArchiveBot
3942:
3920:
3858:
3836:
3402:
3400:
3326:
3293:
3215:
3118:
3085:
3031:
3004:
2956:. The quotient is
2946:
2929:
2880:
2821:
2794:
2755:
2695:
2632:
2596:
2525:
2467:
2420:
2390:
2327:
2261:
2221:
2188:
2150:
2123:
2068:
1964:
1944:
1936:
1928:
1900:
1868:
1809:0,b\geq 0}" /: -->
1799:
1760:
1622:H. Maser (Hrsg.):
1548:
1519:
1465:
1422:
1382:
1363:
1345:
1324:
1278:
1260:
1215:
1191:
1174:
1166:
1122:
1105:
1097:
1069:
840:However, there is
773:quadratic integers
719:
718:
679:
678:
574:
552:
506:gmail <DOT: -->
495:
473:
451:
425:
377:
101:Mathematics portal
45:content assessment
4045:
3908:
3892:
3824:
3808:
3659:more encyclopedic
3344:
3277:
3254:
2985:
2594:
2593:
2523:
2522:
2465:
1967:{\displaystyle b}
1496:
1359:
1274:
1218:{\displaystyle q}
1033:Disputed question
997:
985:comment added by
828:Unsolved problems
716:
713:
676:
673:
620:comment added by
513:comment added by
271:positive integers
227:
226:
166:
165:
162:
161:
158:
157:
4224:
4140:
4138:
4137:
4132:
4130:
4118:
4116:
4115:
4110:
4108:
4079:
4070:
4043:
4042:
4021:
3985:Gaussian integer
3951:
3949:
3948:
3943:
3909:
3901:
3893:
3885:
3867:
3865:
3864:
3859:
3825:
3817:
3809:
3801:
3732:General comments
3521:(2+3i) = 13 and
3411:
3409:
3408:
3403:
3401:
3396:
3395:
3381:
3380:
3359:
3335:
3333:
3332:
3327:
3319:
3318:
3302:
3300:
3299:
3294:
3292:
3278:
3276:
3275:
3260:
3255:
3253:
3252:
3243:
3242:
3233:
3224:
3222:
3221:
3216:
3214:
3194:
3180:
3179:
3170:
3169:
3157:
3156:
3144:
3143:
3127:
3125:
3124:
3119:
3111:
3110:
3094:
3092:
3091:
3086:
3084:
3064:
3040:
3038:
3037:
3032:
3030:
3029:
3013:
3011:
3010:
3005:
3003:
2986:
2984:
2983:
2974:
2973:
2964:
2955:
2953:
2952:
2947:
2939:
2938:
2924:
2910:
2909:
2889:
2887:
2886:
2881:
2879:
2878:
2863:
2862:
2850:
2849:
2830:
2828:
2827:
2822:
2820:
2819:
2803:
2801:
2800:
2795:
2787:
2786:
2764:
2762:
2761:
2756:
2754:
2749:
2748:
2739:
2731:
2726:
2725:
2716:
2704:
2702:
2701:
2696:
2694:
2693:
2681:
2680:
2671:
2670:
2658:
2657:
2641:
2639:
2638:
2633:
2625:
2624:
2605:
2603:
2602:
2597:
2595:
2589:
2588:
2587:
2582:
2581:
2572:
2566:
2561:
2556:
2555:
2546:
2534:
2532:
2531:
2526:
2524:
2518:
2514:
2509:
2505:
2498:
2497:
2476:
2474:
2473:
2468:
2466:
2464:
2463:
2454:
2453:
2444:
2429:
2427:
2426:
2421:
2419:
2418:
2399:
2397:
2396:
2391:
2386:
2381:
2380:
2371:
2363:
2358:
2357:
2348:
2336:
2334:
2333:
2328:
2326:
2325:
2313:
2312:
2303:
2302:
2290:
2289:
2270:
2268:
2267:
2262:
2260:
2259:
2247:
2246:
2230:
2228:
2227:
2222:
2214:
2213:
2197:
2195:
2194:
2189:
2159:
2157:
2156:
2151:
2149:
2148:
2132:
2130:
2129:
2124:
2122:
2121:
2100:
2099:
2077:
2075:
2074:
2069:
2067:
2066:
2054:
2053:
1973:
1971:
1970:
1965:
1953:
1951:
1950:
1945:
1943:
1909:
1907:
1906:
1901:
1877:
1875:
1874:
1869:
1849:
1848:
1810:
1807:
1806:
1800:
1774:0,b\geq 0}": -->
1769:
1767:
1766:
1761:
1583:
1577:
1569:
1563:
1557:
1555:
1554:
1549:
1528:
1526:
1525:
1520:
1502:
1494:
1474:
1472:
1471:
1466:
1452:
1451:
1431:
1429:
1428:
1423:
1418:
1391:
1389:
1388:
1383:
1361:
1360:
1358:
1353:
1348:
1333:
1331:
1330:
1325:
1276:
1275:
1273:
1268:
1263:
1224:
1222:
1221:
1216:
1200:
1198:
1197:
1192:
1190:
1131:
1129:
1128:
1123:
1121:
1078:
1076:
1075:
1070:
1056:
1055:
975:such that X = ΠY
805:commutative ring
729:) with integers
728:
726:
725:
720:
715:
714:
706:
688:
686:
685:
680:
675:
674:
666:
632:
583:
581:
580:
575:
573:
561:
559:
558:
553:
551:
525:
504:
502:
501:
496:
494:
482:
480:
479:
474:
472:
460:
458:
457:
452:
450:
434:
432:
431:
426:
424:
386:
384:
383:
378:
370:
323:
248:Be careful. The
222:
206:
178:
170:
134:
133:
130:
127:
124:
103:
98:
97:
87:
80:
79:
74:
66:
59:
42:
33:
32:
25:
24:
16:
4232:
4231:
4227:
4226:
4225:
4223:
4222:
4221:
4172:
4171:
4121:
4120:
4096:
4095:
4088:
4073:
4068:
4036:
4029:have permission
4019:
3993:this simple FaQ
3978:
3870:
3869:
3786:
3785:
3758:
3590:General omments
3372:
3345:
3338:
3337:
3310:
3305:
3304:
3264:
3244:
3234:
3227:
3226:
3171:
3161:
3148:
3135:
3130:
3129:
3102:
3097:
3096:
3043:
3042:
3021:
3016:
3015:
2975:
2965:
2958:
2957:
2930:
2901:
2896:
2895:
2870:
2854:
2841:
2833:
2832:
2811:
2806:
2805:
2772:
2767:
2766:
2740:
2717:
2707:
2706:
2685:
2672:
2662:
2649:
2644:
2643:
2616:
2611:
2610:
2573:
2567:
2547:
2537:
2536:
2489:
2488:
2484:
2479:
2478:
2455:
2445:
2432:
2431:
2410:
2405:
2404:
2372:
2349:
2339:
2338:
2317:
2304:
2294:
2281:
2276:
2275:
2251:
2238:
2233:
2232:
2205:
2200:
2199:
2162:
2161:
2140:
2135:
2134:
2113:
2091:
2083:
2082:
2058:
2045:
2040:
2039:
2029:Deleted version
2010:
1956:
1955:
1912:
1911:
1883:
1882:
1840:
1820:
1819:
1772:
1771:
1743:
1742:
1663:citation needed
1581:
1579:citation needed
1575:
1565:
1559:
1531:
1530:
1505:
1504:
1493:
1443:
1438:
1437:
1394:
1393:
1336:
1335:
1227:
1226:
1207:
1206:
1147:
1146:
1081:
1080:
1047:
1042:
1041:
1035:
1011:integral domain
978:
974:
968:
966:Gaussian Primes
942:in French mean
830:is as follows:
824:
799:for reverting.
793:
691:
690:
651:
650:
647:
615:
564:
563:
542:
541:
508:
485:
484:
463:
462:
441:
440:
403:
402:
314:
313:
232:
218:
207:
201:
183:
131:
128:
125:
122:
121:
99:
92:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
4230:
4228:
4220:
4219:
4214:
4209:
4204:
4199:
4194:
4189:
4184:
4174:
4173:
4170:
4169:
4143:37.117.118.138
4129:
4107:
4103:
4087:
4084:
4063:
4062:
4055:
4008:
4007:
3999:Added archive
3977:
3974:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3918:
3915:
3912:
3907:
3904:
3899:
3896:
3891:
3888:
3883:
3880:
3877:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3834:
3831:
3828:
3823:
3820:
3815:
3812:
3807:
3804:
3799:
3796:
3793:
3757:
3754:
3753:
3752:
3751:
3750:
3749:
3748:
3747:
3746:
3728:
3706:
3705:
3704:
3703:
3702:
3701:
3662:
3654:
3653:
3652:
3646:
3609:
3587:
3568:
3534:
3451:
3450:
3399:
3394:
3390:
3387:
3384:
3379:
3375:
3371:
3368:
3365:
3362:
3358:
3354:
3351:
3348:
3325:
3322:
3317:
3313:
3291:
3287:
3284:
3281:
3274:
3270:
3267:
3263:
3258:
3251:
3247:
3241:
3237:
3213:
3209:
3206:
3203:
3200:
3197:
3193:
3189:
3186:
3183:
3178:
3174:
3168:
3164:
3160:
3155:
3151:
3147:
3142:
3138:
3117:
3114:
3109:
3105:
3083:
3079:
3076:
3073:
3070:
3067:
3063:
3059:
3056:
3053:
3050:
3028:
3024:
3002:
2998:
2995:
2992:
2989:
2982:
2978:
2972:
2968:
2945:
2942:
2937:
2933:
2927:
2923:
2919:
2916:
2913:
2908:
2904:
2877:
2873:
2869:
2866:
2861:
2857:
2853:
2848:
2844:
2840:
2818:
2814:
2793:
2790:
2785:
2782:
2779:
2775:
2753:
2747:
2743:
2738:
2734:
2730:
2724:
2720:
2715:
2692:
2688:
2684:
2679:
2675:
2669:
2665:
2661:
2656:
2652:
2631:
2628:
2623:
2619:
2592:
2586:
2580:
2576:
2571:
2564:
2560:
2554:
2550:
2545:
2521:
2517:
2512:
2508:
2504:
2501:
2496:
2492:
2487:
2462:
2458:
2452:
2448:
2442:
2439:
2417:
2413:
2401:
2400:
2389:
2385:
2379:
2375:
2370:
2366:
2362:
2356:
2352:
2347:
2324:
2320:
2316:
2311:
2307:
2301:
2297:
2293:
2288:
2284:
2258:
2254:
2250:
2245:
2241:
2220:
2217:
2212:
2208:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2147:
2143:
2120:
2116:
2112:
2109:
2106:
2103:
2098:
2094:
2090:
2065:
2061:
2057:
2052:
2048:
2034:Otherwise the
2022:
2021:
2009:
2006:
2005:
2004:
2003:
2002:
2001:
2000:
1999:
1998:
1997:
1996:
1982:
1979:
1975:
1963:
1942:
1939:
1934:
1931:
1925:
1922:
1919:
1899:
1896:
1893:
1890:
1879:
1867:
1864:
1861:
1858:
1855:
1852:
1847:
1843:
1839:
1836:
1833:
1830:
1827:
1812:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1759:
1756:
1753:
1750:
1739:
1727:
1726:
1725:
1724:
1723:
1722:
1721:
1720:
1701:
1666:
1659:
1648:
1647:
1646:
1645:
1632:
1629:
1620:
1613:
1606:
1597:
1596:
1547:
1544:
1541:
1538:
1518:
1515:
1512:
1478:kind regards--
1464:
1461:
1458:
1455:
1450:
1446:
1421:
1417:
1413:
1410:
1407:
1404:
1401:
1381:
1378:
1375:
1372:
1369:
1366:
1357:
1352:
1343:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1284:
1281:
1272:
1267:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1214:
1189:
1186:
1183:
1180:
1177:
1172:
1169:
1163:
1160:
1157:
1154:
1139:
1138:
1133:
1120:
1117:
1114:
1111:
1108:
1103:
1100:
1094:
1091:
1088:
1068:
1065:
1062:
1059:
1054:
1050:
1034:
1031:
1030:
1029:
976:
972:
967:
964:
963:
962:
961:
960:
959:
958:
931:
930:
929:
928:
911:
910:
896:
887:DISTRIBUTION
842:no explanation
823:
822:Repartition???
820:
797:David Eppstein
792:
789:
788:
787:
754:188.169.229.30
750:
749:
746:
712:
709:
704:
701:
698:
672:
669:
664:
661:
658:
646:
643:
642:
641:
640:
639:
638:
637:
636:
635:
634:
633:
603:
602:
601:
600:
599:
598:
597:
596:
586:David Eppstein
572:
550:
531:
530:
529:
528:
527:
526:
493:
471:
449:
423:
419:
416:
413:
410:
394:
393:
392:
391:
390:
389:
388:
387:
376:
373:
369:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
322:
304:
303:
302:
301:
300:
299:
291:
290:
289:
288:
260:
259:
250:imaginary part
231:
228:
225:
224:
212:
209:
208:
203:
199:
197:
194:
193:
185:
184:
179:
173:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
4229:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4179:
4177:
4168:
4164:
4160:
4155:
4154:
4153:
4152:
4148:
4144:
4101:
4093:
4085:
4083:
4082:
4077:
4072:
4071:
4060:
4056:
4053:
4049:
4048:
4047:
4040:
4034:
4030:
4026:
4022:
4016:
4011:
4006:
4002:
3998:
3997:
3996:
3994:
3990:
3986:
3981:
3975:
3973:
3972:
3968:
3964:
3960:
3955:
3953:
3936:
3933:
3927:
3924:
3921:
3916:
3913:
3910:
3905:
3902:
3897:
3894:
3889:
3886:
3881:
3878:
3875:
3852:
3849:
3843:
3840:
3837:
3832:
3829:
3826:
3821:
3818:
3813:
3810:
3805:
3802:
3797:
3794:
3791:
3781:
3777:
3773:
3769:
3767:
3766:Quotient ring
3762:
3755:
3745:
3741:
3737:
3733:
3729:
3727:
3723:
3719:
3714:
3713:
3712:
3711:
3710:
3709:
3708:
3707:
3700:
3696:
3692:
3688:
3684:
3680:
3679:
3678:
3674:
3670:
3666:
3663:
3660:
3655:
3650:
3647:
3644:
3641:
3640:
3639:
3636:
3635:
3634:
3630:
3626:
3622:
3618:
3614:
3610:
3607:
3603:
3599:
3595:
3591:
3588:
3586:
3582:
3578:
3574:
3569:
3567:
3563:
3559:
3554:
3550:
3548:
3544:
3540:
3535:
3532:
3528:
3524:
3520:
3516:
3512:
3508:
3505:divides also
3504:
3500:
3496:
3494:
3490:
3485:
3484:
3483:
3482:
3478:
3474:
3469:
3466:
3464:
3460:
3456:
3449:
3445:
3444:
3443:
3441:
3437:
3433:
3429:
3424:
3423:
3418:
3417:
3413:
3397:
3388:
3385:
3382:
3377:
3373:
3369:
3363:
3360:
3352:
3349:
3323:
3320:
3315:
3311:
3285:
3282:
3279:
3268:
3265:
3261:
3256:
3249:
3245:
3239:
3235:
3207:
3204:
3201:
3198:
3195:
3187:
3184:
3181:
3176:
3172:
3166:
3162:
3158:
3153:
3149:
3145:
3140:
3136:
3115:
3112:
3107:
3103:
3077:
3074:
3071:
3068:
3065:
3057:
3054:
3051:
3048:
3026:
3022:
2996:
2993:
2990:
2987:
2980:
2976:
2970:
2966:
2943:
2940:
2935:
2931:
2925:
2917:
2914:
2911:
2906:
2902:
2893:
2875:
2871:
2867:
2859:
2855:
2851:
2846:
2842:
2816:
2812:
2791:
2788:
2783:
2780:
2777:
2773:
2745:
2741:
2732:
2722:
2718:
2690:
2686:
2682:
2677:
2673:
2667:
2663:
2659:
2654:
2650:
2629:
2626:
2621:
2617:
2607:
2590:
2578:
2574:
2562:
2552:
2548:
2519:
2515:
2510:
2506:
2502:
2499:
2494:
2490:
2485:
2460:
2456:
2450:
2446:
2440:
2437:
2415:
2411:
2387:
2377:
2373:
2364:
2354:
2350:
2322:
2318:
2314:
2309:
2305:
2299:
2295:
2291:
2286:
2282:
2274:
2273:
2272:
2256:
2252:
2248:
2243:
2239:
2218:
2215:
2210:
2206:
2185:
2182:
2176:
2173:
2170:
2145:
2141:
2118:
2114:
2110:
2104:
2101:
2096:
2092:
2079:
2063:
2059:
2055:
2050:
2046:
2037:
2032:
2030:
2026:
2020:
2016:
2015:
2014:
2007:
1995:
1991:
1987:
1983:
1980:
1976:
1961:
1937:
1932:
1923:
1920:
1917:
1897:
1894:
1891:
1888:
1880:
1865:
1862:
1859:
1856:
1853:
1850:
1845:
1837:
1834:
1831:
1828:
1817:
1813:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1757:
1754:
1751:
1748:
1740:
1737:
1736:
1735:
1734:
1733:
1732:
1731:
1730:
1729:
1728:
1719:
1715:
1711:
1707:
1702:
1698:
1694:
1690:
1686:
1681:
1680:
1679:
1675:
1671:
1667:
1664:
1660:
1658:
1654:
1653:
1652:
1651:
1650:
1649:
1644:
1640:
1636:
1633:
1630:
1627:
1626:
1621:
1618:
1614:
1611:
1607:
1605:
1601:
1600:
1599:
1598:
1595:
1591:
1587:
1580:
1573:
1568:
1562:
1545:
1542:
1539:
1536:
1516:
1513:
1510:
1500:
1492:
1491:
1490:
1489:
1485:
1481:
1477:
1462:
1459:
1456:
1453:
1448:
1444:
1435:
1419:
1408:
1405:
1402:
1376:
1373:
1370:
1364:
1355:
1350:
1341:
1318:
1315:
1312:
1306:
1303:
1300:
1294:
1291:
1288:
1285:
1279:
1270:
1265:
1256:
1253:
1250:
1244:
1241:
1235:
1232:
1212:
1204:
1184:
1181:
1178:
1175:
1170:
1161:
1158:
1155:
1152:
1144:
1137:
1134:
1132:
1115:
1112:
1109:
1106:
1101:
1092:
1089:
1086:
1066:
1063:
1060:
1057:
1052:
1048:
1037:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1004:
1003:prime element
1000:
999:
998:
996:
992:
988:
987:74.96.218.225
984:
965:
957:
953:
949:
945:
941:
937:
936:
935:
934:
933:
932:
927:
923:
919:
915:
914:
913:
912:
909:
905:
901:
897:
894:
890:
886:
883:
880:
877:
876:
875:
874:
870:
866:
861:
856:
853:
851:
846:
843:
838:
836:
831:
829:
821:
819:
818:
814:
810:
806:
800:
798:
790:
786:
782:
778:
774:
770:
766:
765:
764:
763:
759:
755:
747:
744:
743:
742:
740:
736:
732:
710:
707:
702:
699:
696:
670:
667:
662:
659:
656:
644:
631:
627:
623:
622:125.63.107.34
619:
613:
612:
611:
610:
609:
608:
607:
606:
605:
604:
595:
591:
587:
539:
538:
537:
536:
535:
534:
533:
532:
524:
520:
516:
512:
438:
417:
414:
411:
408:
400:
399:
398:
397:
396:
395:
374:
363:
360:
357:
354:
351:
348:
345:
342:
339:
333:
327:
312:
311:
310:
309:
308:
307:
306:
305:
297:
296:
295:
294:
293:
292:
287:
284:
280:
276:
272:
268:
264:
263:
262:
261:
258:
255:
251:
247:
246:
245:
244:
241:
237:
229:
221:
216:
211:
210:
196:
195:
192:
191:
187:
186:
182:
177:
172:
171:
168:
153:
149:
148:High-priority
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
73:High‑priority
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
4089:
4067:
4064:
4039:source check
4018:
4012:
4009:
3982:
3979:
3956:
3770:
3759:
3731:
3681:Please read
3664:
3658:
3648:
3642:
3638:My comments:
3637:
3602:WP:TECHNICAL
3597:
3589:
3552:
3546:
3542:
3538:
3530:
3526:
3522:
3518:
3514:
3510:
3506:
3502:
3498:
3492:
3488:
3470:
3467:
3462:
3458:
3454:
3452:
3447:
3439:
3435:
3431:
3427:
3425:
3420:
3419:
3415:
3414:
2891:
2608:
2402:
2160:(especially
2080:
2033:
2028:
2027:
2023:
2018:
2011:
1815:
1696:
1692:
1688:
1684:
1662:
1623:
1616:
1609:
1566:
1560:
1475:
1433:
1202:
1142:
1140:
1135:
1039:
981:— Preceding
969:
944:distribution
943:
939:
892:
888:
884:
881:
878:
859:
857:
854:
849:
847:
841:
839:
834:
832:
827:
825:
801:
794:
777:Mark Dominus
751:
738:
734:
730:
648:
616:— Preceding
515:125.63.107.5
509:— Preceding
436:
233:
214:
188:
180:
167:
147:
107:
51:WikiProjects
34:
3780:WP:Civility
3683:WP:Civility
3545:instead of
3491:instead of
3461:instead of
3453:He claims:
3430:instead of
2198:). And for
1038:Quotation:
1005:is also an
948:Mark viking
940:Répartition
879:repartition
809:Prim Ethics
123:Mathematics
114:mathematics
70:Mathematics
4176:Categories
4076:Report bug
1803:0,b\geq 0}
401:Shouldn't
230:Definition
4059:this tool
4052:this tool
3592:: Before
3517:(3+2i) =
2081:It holds
1978:insights.
918:Anita5192
900:Anita5192
269:with the
39:is rated
4159:D.Lazard
4065:Cheers.—
3963:D.Lazard
3772:Wolfk.wk
3761:Wolfk.wk
3736:Wolfk.wk
3718:Wolfk.wk
3691:D.Lazard
3669:Wolfk.wk
3625:D.Lazard
3621:Wolfk.wk
3613:Wolfk.wk
3606:Wolfk.wk
3594:Wolfk.wk
3577:D.Lazard
3573:Wolfk.wk
3558:D.Lazard
3493:N(a + b)
3473:Wolfk.wk
3446:Section
3432:N(a + b)
3128:and get
2892:Example:
2133:for all
2017:Section
1986:Wolfk.wk
1710:D.Lazard
1706:lattices
1670:Wolfk.wk
1657:D.Lazard
1635:Wolfk.wk
1617:page 546
1604:D.Lazard
1586:D.Lazard
1499:Wolfk.wk
1480:Wolfk.wk
1225:, with
1019:D.Lazard
983:unsigned
752:Thanks!
618:unsigned
511:unsigned
437:integers
279:argument
267:integers
254:Dmharvey
240:JackofOz
236:integers
215:365 days
181:Archives
3989:my edit
1700:remark.
1691:, with
1497:editor
1334:, i.e.
863:used!!!
275:modulus
150:on the
41:C-class
3959:WT:WPM
3438:) but
3014:. For
1145:holds
1009:in an
850:please
562:, but
47:scale.
4086:Wrong
3868:and
3598:after
3489:N(ab)
3428:N(ab)
2271:with
1974:even.
1910:with
1782:: -->
1770:with
1572:WP:OR
1558:with
895:1969)
737:(and
507:com
277:with
28:This
4163:talk
4147:talk
3967:talk
3740:talk
3722:talk
3695:talk
3673:talk
3629:talk
3581:talk
3562:talk
3553:this
3543:N(m)
3501:and
3477:talk
3459:N(m)
3442:!!
2733:<
2705:and
2365:<
2337:and
1990:talk
1954:and
1714:talk
1674:talk
1639:talk
1590:talk
1484:talk
1023:talk
991:talk
952:talk
922:talk
904:talk
869:talk
865:Daqu
860:also
813:talk
781:talk
771:and
758:talk
733:and
626:talk
590:talk
519:talk
283:PhiJ
142:High
4119:in
4033:RfC
4003:to
3643:GCD
3611:So
3495:...
2997:0.5
2991:2.5
2609:If
1933:mod
1655:To
1602:To
1171:mod
1102:mod
4178::
4165:)
4149:)
4046:.
4041:}}
4037:{{
3969:)
3940:∞
3937:…
3931:∞
3928:−
3914:…
3898:±
3882:±
3856:∞
3853:…
3847:∞
3844:−
3830:…
3814:±
3798:±
3742:)
3724:)
3716:--
3697:)
3675:)
3667:--
3631:)
3583:)
3564:)
3509:+
3479:)
3471:--
3412:.
3398:_
3286:−
3196:−
3159:−
2890:.
2627:≠
2606:.
2563:≤
2511:≤
2503:ξ
2500:−
2441::=
2438:ξ
2216:≠
2031::
1992:)
1921:≡
1854:−
1794:≥
1716:)
1689:ib
1676:)
1668:--
1641:)
1592:)
1584:.
1582:}}
1576:{{
1495:To
1486:)
1242:−
1236:−
1159:−
1156:≡
1090:≡
1058::=
1025:)
993:)
954:)
924:)
906:)
889:2:
885:1:
871:)
837:"
815:)
783:)
760:)
708:−
668:−
628:)
592:)
521:)
418:∈
364:∈
352:∣
4161:(
4145:(
4128:Z
4106:Z
4102:2
4078:)
4074:(
4061:.
4054:.
3965:(
3952:"
3934:,
3925:=
3922:s
3917:,
3911:,
3906:2
3903:3
3895:,
3890:2
3887:1
3879:=
3876:t
3850:,
3841:=
3838:t
3833:,
3827:,
3822:2
3819:3
3811:,
3806:2
3803:1
3795:=
3792:s
3738:(
3720:(
3693:(
3671:(
3661:.
3627:(
3579:(
3560:(
3547:m
3537:"
3531:b
3529:+
3527:a
3525:(
3523:N
3519:N
3515:N
3511:b
3507:a
3503:b
3499:a
3475:(
3463:m
3440:1
3393:i
3389:+
3386:1
3383:=
3378:2
3374:z
3370:=
3367:)
3364:2
3361:,
3357:i
3353:+
3350:5
3347:(
3324:0
3321:=
3316:3
3312:z
3290:i
3283:1
3280:=
3273:i
3269:+
3266:1
3262:2
3257:=
3250:2
3246:z
3240:1
3236:z
3212:i
3208:+
3205:1
3202:=
3199:4
3192:i
3188:+
3185:5
3182:=
3177:1
3173:z
3167:1
3163:q
3154:0
3150:z
3146:=
3141:2
3137:z
3116:2
3113:=
3108:1
3104:q
3082:i
3078:+
3075:3
3072:,
3069:3
3066:,
3062:i
3058:+
3055:2
3052:,
3049:2
3027:1
3023:q
3001:i
2994:+
2988:=
2981:1
2977:z
2971:0
2967:z
2944:2
2941:=
2936:1
2932:z
2926:,
2922:i
2918:+
2915:5
2912:=
2907:0
2903:z
2876:n
2872:z
2868:=
2865:)
2860:1
2856:z
2852:,
2847:0
2843:z
2839:(
2817:n
2813:z
2792:0
2789:=
2784:1
2781:+
2778:n
2774:z
2752:|
2746:2
2742:z
2737:|
2729:|
2723:3
2719:z
2714:|
2691:3
2687:z
2683:+
2678:2
2674:z
2668:2
2664:q
2660:=
2655:1
2651:z
2630:0
2622:2
2618:z
2591:2
2585:|
2579:1
2575:z
2570:|
2559:|
2553:2
2549:z
2544:|
2520:2
2516:1
2507:|
2495:1
2491:q
2486:|
2461:1
2457:z
2451:0
2447:z
2416:1
2412:q
2388:.
2384:|
2378:1
2374:z
2369:|
2361:|
2355:2
2351:z
2346:|
2323:2
2319:z
2315:+
2310:1
2306:z
2300:1
2296:q
2292:=
2287:0
2283:z
2257:2
2253:z
2249:,
2244:1
2240:q
2219:0
2211:1
2207:z
2186:0
2183:=
2180:)
2177:0
2174:,
2171:0
2168:(
2146:0
2142:z
2119:0
2115:z
2111:=
2108:)
2105:0
2102:,
2097:0
2093:z
2089:(
2064:1
2060:z
2056:,
2051:0
2047:z
1988:(
1962:b
1941:)
1938:4
1930:(
1924:1
1918:a
1898:i
1895:b
1892:+
1889:a
1878:.
1866:i
1863:4
1860:+
1857:3
1851:=
1846:2
1842:)
1838:i
1835:2
1832:+
1829:1
1826:(
1811:.
1797:0
1791:b
1788:,
1785:0
1779:a
1758:i
1755:b
1752:+
1749:a
1712:(
1697:b
1693:a
1687:+
1685:a
1672:(
1665:.
1637:(
1612:.
1588:(
1567:b
1561:a
1546:b
1543:i
1540:+
1537:a
1517:i
1514:+
1511:1
1501::
1482:(
1463:i
1460:+
1457:1
1454:=
1449:1
1445:p
1420:z
1416:|
1412:)
1409:i
1406:+
1403:1
1400:(
1380:)
1377:i
1374:+
1371:1
1368:(
1365:q
1356:!
1351:=
1342:z
1322:)
1319:i
1316:+
1313:1
1310:(
1307:q
1304:2
1301:=
1298:)
1295:i
1292:2
1289:+
1286:2
1283:(
1280:q
1271:!
1266:=
1257:z
1254:2
1251:=
1248:)
1245:z
1239:(
1233:z
1213:q
1188:)
1185:i
1182:2
1179:+
1176:2
1168:(
1162:z
1153:z
1119:)
1116:i
1113:2
1110:+
1107:2
1099:(
1093:1
1087:p
1067:i
1064:+
1061:1
1053:1
1049:p
1021:(
989:(
977:i
973:i
950:(
920:(
902:(
882:n
867:(
833:"
811:(
779:(
756:(
739:k
735:b
731:a
711:k
703:b
700:+
697:a
671:2
663:b
660:+
657:a
624:(
588:(
571:Z
549:I
517:(
492:Z
470:I
448:R
422:I
415:b
412:,
409:a
375:.
372:}
368:Z
361:b
358:,
355:a
349:i
346:b
343:+
340:a
337:{
334:=
331:]
328:i
325:[
321:Z
190:1
154:.
53::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
