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Also, there are some cases, such as where we have a Cauchy-distributed random variable, where the average value does not exist mathematically, but there is a "typical" value that can be defined. It might be a median, or a value of maximum likelihood, or to get more technical, it could be the "Cauchy
188:
In many mathematical, statistical, and scientific, and engineering applications, the average value and the typical value are one and the same. (So, there would be no reason to mention both of them.) On the other hand, there are situations where the average value exists, but some other value (such as
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In the phrase "for every integer a that is both coprime to and smaller than n." the limitation "smaller than n." doesn't seem to be necessary. a^m mod n = 1 is valid for any integer a coprime to n, according to at least one other source. I ask a math expert to verify this, I don't know whether this
214:
I think the recursive definition of the function is incomplete. What to do with the cases p = 2^1 and p = 2^2 ? The example sequence (1,1,2,2,...) indicates these ones fall in the general p^(k-1)*(p-1) case. I'm no mathematician, could a math guy (or girl) fix this?
1281:
In the first paragraph of numerical examples, after 2^2=4\ne 1 (mod 5), the fact that 3^2=9\ne 1 (mod 5) is irrelevant, it should read 2^3=8\ne 1 (mod 5), as we want to show that the 4th is the minimal power, as 2nd and 3rd powers do not result in 1.
192:
Principal Value" of the average. Note that the Cauchy
Principal Value of something is not just for Cauchy random variables, but it is a more general concept that can be applied to lots of infinite sums and indefinite integrals.
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The answers.com link refs wikipedia as the source so their info came from here rather than the other way around. I do find the article very difficult to follow though so a rewrite might be a good idea.
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is in parentheses suggests to me that it was considered additional reinforcement, not necessary to make the point, given that exponent 2 was already ruled out by the example
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And then there is the section that is called "Average and
Typical value", but with no explanation of what these really mean, and what is the distinction between the two.
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1411:. I suspect that exponent 3 was not mentioned since it's ruled out on the general principle that the exponent of a group must divide the order of the group, which is
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I have made some changes, following up on my previous comment. Let me know if you think anything important or clarifying has been omitted.
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Values for 15 and 16 are shown in bold indicating they are different when the actual values shown seem to match. --
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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is a mistake or simply a more recent result not included in the original definition of the
Carmichael function.
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I'm afraid that Lower Bound in
Theorem 1 makes very little sense. Are you sure that everything is perfectly OK?
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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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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before doing mass systematic removals. This message is updated dynamically through the template
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https://web.archive.org/web/20110615120521/http://www25.brinkster.com/denshade/totient.html
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If you found an error with any archives or the URLs themselves, you can fix them with
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in this example. Might be worth adding a remark about that to the introduction.
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And why this page is - well - extremele similar to that on answers.com?
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When you have finished reviewing my changes, please set the
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for additional information. I made the following changes:
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749:{\displaystyle a^{2}\equiv 9\equiv 1{\text{ mod 4}}}
101:, a collaborative effort to improve the coverage of
1192:using the archive tool instructions below. Editors
865:{\displaystyle a^{\psi (4)}\equiv 1{\text{ mod 4}}}
493:{\displaystyle a^{\psi (2)}\equiv 1{\text{ mod 2}}}
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210:recursive definition of the function is incomplete
1150:http://www25.brinkster.com/denshade/totient.html
167:http://www.answers.com/topic/carmichael-function
158:Lower Bound in Theorem 1 makes very little sense
1404:{\displaystyle 2^{2}=4\not \equiv 1{\pmod {5}}}
1347:{\displaystyle 3^{2}=9\not \equiv 1{\pmod {5}}}
1178:This message was posted before February 2018.
189:possibly some median value) is more "typical".
8:
705:{\displaystyle a^{1}\equiv 3{\text{ mod 4}}}
667:{\displaystyle a^{1}\equiv 1{\text{ mod 4}}}
377:{\displaystyle a^{1}\equiv 1{\text{ mod 2}}}
1128:I have just modified one external link on
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1086:{\displaystyle a^{m}\equiv 1{\pmod {n}}}
1021:{\displaystyle a^{m}\equiv 1{\pmod {n}}}
956:{\displaystyle a^{m}\equiv 1{\pmod {n}}}
756:, where the Euler's totient function of
384:, where the Euler's totient function of
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1167:to let others know (documentation at
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95:This article is within the scope of
890:phrase doesn't seem to be necessary
38:It is of interest to the following
1251:Table of first values inconsistent
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1492:Low-priority mathematics articles
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1132:. Please take a moment to review
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115:Knowledge:WikiProject Mathematics
1264:Nevermind. I see what I did. --
118:Template:WikiProject Mathematics
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1439:{\displaystyle \varphi (5)=4}
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1269:03:24, 1 September 2018 (UTC)
1246:21:23, 15 November 2016 (UTC)
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346:. One can immediately obtain
109:and see a list of open tasks.
1487:C-Class mathematics articles
882:04:24, 15 January 2018 (UTC)
178:11:59, 11 October 2006 (UTC)
1472:16:10, 16 August 2024 (UTC)
1456:16:53, 14 August 2024 (UTC)
1292:09:41, 14 August 2024 (UTC)
1260:08:06, 31 August 2018 (UTC)
1093:for every positive integer
546:be coprime with respect to
500:. Likewise, in the case of
291:be coprime with respect to
202:15:58, 6 October 2008 (UTC)
1508:
1209:(last update: 5 June 2024)
1125:Hello fellow Wikipedians,
1116:13:13, 19 March 2018 (UTC)
905:14:18, 10 March 2010 (UTC)
236:11:15, 10 March 2010 (UTC)
134:
67:
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798:{\displaystyle \psi (4)}
426:{\displaystyle \psi (2)}
141:project's priority scale
1121:External links modified
98:WikiProject Mathematics
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339:{\displaystyle a=1+2n}
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1159:parameter below to
1130:Carmichael function
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519:{\displaystyle p=4}
264:{\displaystyle p=2}
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1277:Numerical examples
1234:InternetArchiveBot
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34:content assessment
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818:{\displaystyle 2}
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559:{\displaystyle p}
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304:{\displaystyle p}
284:{\displaystyle a}
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222:comment added by
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837:
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825:. Therefore,
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137:Low-priority
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62:Low‑priority
40:WikiProjects
1464:Will Orrick
1448:Will Orrick
1171:Sourcecheck
311:; that is,
218:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
1481:Categories
1241:Report bug
1224:this tool
1217:this tool
1108:AxelBoldt
1230:Cheers.—
897:Rkomatsu
453:. Thus,
232:contribs
224:Rkomatsu
220:unsigned
1157:checked
1134:my edit
139:on the
30:C-class
1165:failed
874:D4nn0v
712:, and
526:, let
271:, let
36:scale.
1297:That
1284:LaurV
859:mod 4
743:mod 4
699:mod 4
661:mod 4
487:mod 2
371:mod 2
1468:talk
1452:talk
1288:talk
1161:true
1112:talk
1040:and
975:and
901:talk
878:talk
228:talk
198:talk
1390:mod
1333:mod
1198:RfC
1175:).
1163:or
1148:to
1072:mod
1007:mod
942:mod
601:or
131:Low
1483::
1470:)
1454:)
1419:φ
1290:)
1266:N8
1257:N8
1211:.
1206:}}
1202:{{
1173:}}
1169:{{
1114:)
1060:≡
995:≡
930:≡
903:)
880:)
872:.
852:≡
838:ψ
784:ψ
776:,
736:≡
730:≡
692:≡
654:≡
480:≡
466:ψ
412:ψ
404:,
364:≡
234:)
230:•
200:)
1466:(
1450:(
1434:4
1431:=
1428:)
1425:5
1422:(
1398:)
1395:5
1387:(
1381:1
1378:≢
1375:4
1372:=
1367:2
1363:2
1341:)
1338:5
1330:(
1324:1
1321:≢
1318:9
1315:=
1310:2
1306:3
1286:(
1243:)
1239:(
1226:.
1219:.
1110:(
1099:n
1095:a
1080:)
1077:n
1069:(
1063:1
1055:m
1051:a
1034:n
1030:a
1015:)
1012:n
1004:(
998:1
990:m
986:a
969:n
965:a
950:)
947:n
939:(
933:1
925:m
921:a
899:(
876:(
855:1
847:)
844:4
841:(
834:a
813:2
793:)
790:4
787:(
764:4
739:1
733:9
725:2
721:a
695:3
687:1
683:a
657:1
649:1
645:a
624:n
621:4
618:+
615:3
612:=
609:a
589:n
586:4
583:+
580:1
577:=
574:a
554:p
534:a
514:4
511:=
508:p
483:1
475:)
472:2
469:(
462:a
441:1
421:)
418:2
415:(
392:2
367:1
359:1
355:a
334:n
331:2
328:+
325:1
322:=
319:a
299:p
279:a
259:2
256:=
253:p
226:(
196:(
143:.
42::
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