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This is true. Nevertheless, this article lacks of section "Modular operations", in which modular operations (modular addition, modular multiplication, modular exponentiation, modular inverse, and modular division) are defined and described. For the moment this is only partially done, and distributed
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If b has a multiplive inverse mod n (b has a multiplive inverse mod n if and only if gcd(b,n)=1), then we can define (a/b) in arithmetic mod n, the definition is a × (the multiplive inverse of b mod n), e.g. in arithmetic mod 35, 1/2 is 18 and 1/3 is 12, but 1/5 is undefined, since 5 has no
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as no evidence is provided that 1 is considered as a possible modulus in standard textbooks. Also, such a change requires to verify that all properties listed in the article remain correct after the change. This has clearly not been done, as one of the properties begins with "If
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Indeed it doesn't work. Copying and pasting both
Anononymous' code and Knowledge's article both fail with integers close to ULLONG_MAX of clmits, in particular (ULLONG_MAX-1)^2 modulo (ULLONG_MAX-82). It is better to just use the property: (a*b) mod m = (a-m)*(b-m) mod m.
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I have seen this problem elsewhere from time to time on
Knowledge. An IP tried to fix it by inserting spaces, to no avail. I removed the spaces, because they did not help. The only thing that worked for me was to increase the magnification on my
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615:
The source code in the page (reproduced below) allows values for m as large as 2^63 - 1, while simpler source codes usually require that m < 2^32, because of intermediate squaring operations involved that would overflow a 64-bits integer.
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about my supposed understanding of mathematics are not an argument in this discussion. On the opposite, they weaken your position. Please remove them. If you remove the preceding post I would agree that you remove also my answer.
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This was reverted because it is too complicated to be useful in this article. In fact, it is too complicated for most people to verify. If you want to reinsert it, then first discuss it here and demonstrate why it is
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As congruences modulo 0 and 1 are commonly considered (even in this article), it seems that 0 and 1 are rarely considered as moduli. So I suggest to change the beginning of the section into
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And I could not find one(¬1) basic property listed in the article which becomes nonsensical. I asked him to show me one, if not all, of the listed basic properties which become nonsensical.
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Isn't it kind of silly to refer to the 24-hour clock as "military" time since it's the standard timekeeping format in most of the world? Seems anglocentric and an unnecessary parenthetical.
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Good point. I live in a county usually using 12-hour time and it still feels like an
Americanism to me. I've reworded things for now, if anyone disagrees we can go to the third stage of
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rejects the modulus 1 with the argument: "1 is never used as a modulus, and extending the definition to would make nonsensical some of the listed basic properties listed below".
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is lacking, although fundamental. Also fundamental and lacking are: the use of modular arithmetic for efficient linear algebra over the rationals, and the difficult problem of
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The article should mention the fact that full modular arithmetic cannot be extended to reals; and there should be a separate article about this partial arithmetic.
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Indeed, I translated it in C# as follows and I don't get correct results (tested against the BigInteger.ModPow(A, B, M) function provided in the .Net version 4):
2677:. I don't propose to directly affect this page, but this page is listed upon that disambiguation page. Please visit the discussion there and contribute! Thanks —
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What about division? In arithmetic mod 13, 1/2 is 7 since 7*2=1. In arithmetic mod 10, 1/2 is impossible. With prime modulo, only n/0 is impossible.
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The source code below requires that M < 2^32, but at least it gives the correct result (tested against the BigInteger.ModPow(A, B, M) function):
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I agree that the change does not carry much of a fluidum. But it is correct and even useful at least e.g. for certain generic theorems.
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I agree that congruences may and should be defined modulo any nonnegative integer, but this does implies that 0 and 1 may be called
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2263:{\displaystyle {\begin{aligned}-8&\equiv 7{\pmod {5}}\\2&\equiv -3{\pmod {5}}\\-3&\equiv -8{\pmod {5}}.\end{aligned}}}
561:. In other words, this article needs a complete rewriting. I intended to do that, but I have not yet get the time for doing this.
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This code here might be easier to understand, and the loop won't always run 64 iterations. It works for any uint64_t.
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So the change does not improve the article, although the first line of the section requires some attention.
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The first minus sign in the following equations in the
Examples section does not seem to render properly.
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2623:{\displaystyle 2x\equiv 14{\pmod {13}}\equiv 1{\pmod {13}}\iff x\equiv {\frac {1}{2}}{\pmod {13}}}
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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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By the way, the integers modulo 1 are more than the "1-elementic trivial group". They form the
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These operations are reasonably well behaved; if one defines the typical element of
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Can someone help? Otherwise the source code above should be removed from the page.
2854:{\displaystyle \mathbb {Z} /1\mathbb {Z} =\mathbb {Z} /1=\mathbb {Z} _{1}=\{0\}}
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Clearly, such a change would require an update of the article for testing when
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Some use "equivalent to," although I suppose this may be colloquial in usage.
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in several sections. The fact that modular inverses may be computed by either
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Hopefully, the following I added won't be reversed and deleted in future. If
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One cannot define full arithmetic (including multiplication) for the set
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This is happening in several articles. I just put in a help request at
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This artithmetic has many applications, such as computing with angles (
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However, I would add a statement to the paragraph "Integers modulo
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must be explicitly excluded. (This is implicily excluded when
2031:=2π), doing analysis and analytic geometry on a "flat torus" (
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3134:. This is the motivation of the above suggested formulation.
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Hi all! I have started a discussion for the primary topic of
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Knowledge:Village pump (proposals)#Minus signs not rendering
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Does the glyph for congruence about a modulus have a name?
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Talk:Modulo (mathematics)#Requested move 28 December 2022
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Two sources cited in the article define congruence for
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I tested it out on C, and it appears to work for me.
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46:for general discussion of the article's subject.
3322:Knowledge level-4 vital articles in Mathematics
2691:If not, I personally propose the “Threequals.”
2935:in the definition given in the first line of
467:This page has archives. Sections older than
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8:
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2047:=1), computing atom distances in crystals (
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2724:Stupid name. Threequals is way better.
2078:2604:2D80:D686:1800:957E:4D42:301E:9A32
477:when more than 10 sections are present.
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3087:is supposed to be prime or composite.)
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496:2A01:119F:2E9:2F00:94F9:AF78:6BF0:839F
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3327:C-Class vital articles in Mathematics
2726:2001:56A:FCFE:E200:C09:49E7:BAD5:1973
2444:{\displaystyle x\equiv 7{\pmod {13}}}
7:
339:This article is within the scope of
282:It is of interest to the following
36:for discussing improvements to the
2863:is the 1-elementic trivial group.
1950:numbers modulo some positive real
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3337:Top-priority mathematics articles
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1956:However, one can define addition
471:may be automatically archived by
359:Knowledge:WikiProject Mathematics
3307:Knowledge level-4 vital articles
3094:. I'll ass this to the article.
2309:Recently reverted text insertion
1988:) of an ordinary integer number
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362:Template:WikiProject Mathematics
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58:Click here to start a new topic.
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353:and see a list of open tasks.
55:Put new text under old text.
3332:C-Class mathematics articles
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3183:18:50, 5 February 2024 (UTC)
3165:10:22, 5 February 2024 (UTC)
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2720:13:57, 2 December 2023 (UTC)
2701:07:51, 2 December 2023 (UTC)
2511:{\displaystyle \gcd(2,13)=1}
1932:19:59, 30 January 2019 (UTC)
551:extended Euclidean algorithm
3187:Clearer — and critics wrt.
2051:= crystal cell size), etc..
584:multiplive inverse mod 35.
571:21:36, 31 August 2016 (UTC)
544:18:15, 31 August 2016 (UTC)
504:16:59, 31 August 2016 (UTC)
63:New to Knowledge? Welcome!
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2748:06:27, 15 April 2024 (UTC)
2656:16:04, 24 March 2022 (UTC)
2640:15:56, 24 March 2022 (UTC)
2065:01:50, 14 April 2019 (UTC)
559:discrete modular logarithm
3259:Elements of Number Theory
3024:, if there is an integer
2115:Minus signs not rendering
2021:in ℤ }, then ± = , and
600:11:03, 27 July 2018 (UTC)
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93:Be welcoming to newcomers
22:Skip to table of contents
3229:Long, Calvin T. (1972).
2990:Euler's totient function
2966:Euler's totient function
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385:project's priority scale
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2928:{\displaystyle n\geq 1}
2671:Modulo (disambiguation)
2664:Modulo (disambiguation)
2109:07:21, 8 May 2020 (UTC)
2086:02:23, 8 May 2020 (UTC)
1045:// 0x8000000000000000UL
555:Fermat's little theorem
342:WikiProject Mathematics
3302:C-Class vital articles
3233:(2nd ed.). Lexington:
3191:'s emotions removed. -
3151:mixes up "0 and 1" as
3116:any positive integer.—
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2972:= 2 and modulo 0 when
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1764:// if (sum + a) < m
474:Lowercase sigmabot III
88:avoid personal attacks
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113:Neutral point of view
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611:Source code correct?
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118:No original research
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3003:nonnegative integer
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2470:{\displaystyle k=2}
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2222:
2213:
2212:
2181:
2175:
2174:
2146:
2131:
2130:
2119:
2117:
2072:
2070:"Military" time
1940:
1913:
1892:
1891:
1888:
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1035:
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1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
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999:
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993:
990:
987:
984:
981:
978:
975:
972:
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966:
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947:
944:
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914:
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827:
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788:
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776:
773:
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767:
764:
761:
758:
755:
752:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
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714:
711:
708:
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689:
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677:
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665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
613:
486:
472:
461:
455:
416:
364:
361:
358:
355:
354:
332:
325:
305:
276:on Knowledge's
273:
263:
243:
242:
237:
214:
134:
129:
128:
127:
104:
74:
12:
11:
5:
3350:
3348:
3340:
3339:
3334:
3329:
3324:
3319:
3314:
3309:
3304:
3294:
3293:
3287:
3286:
3272:
3247:
3220:
3219:
3215:
3214:
3213:
3212:
3211:
3210:
3209:
3208:
3207:
3206:
3205:
3204:
3203:
3107:
3106:
3088:
3074:
2997:
2979:
2978:
2977:
2924:
2921:
2918:
2896:
2893:
2890:
2850:
2847:
2844:
2841:
2836:
2831:
2826:
2823:
2819:
2814:
2810:
2806:
2802:
2798:
2793:
2783:
2762:
2756:
2755:
2754:
2753:
2752:
2751:
2750:
2740:67.170.223.108
2688:
2685:
2666:
2660:
2659:
2658:
2618:
2615:
2610:
2607:
2599:
2596:
2591:
2588:
2583:
2577:
2574:
2569:
2566:
2560:
2557:
2553:
2550:
2545:
2542:
2536:
2533:
2530:
2527:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2466:
2463:
2460:
2439:
2436:
2431:
2428:
2422:
2419:
2416:
2310:
2307:
2306:
2305:
2271:
2270:
2255:
2251:
2248:
2243:
2240:
2234:
2231:
2228:
2225:
2223:
2221:
2218:
2215:
2214:
2210:
2207:
2202:
2199:
2193:
2190:
2187:
2184:
2182:
2180:
2177:
2176:
2172:
2169:
2164:
2161:
2155:
2152:
2149:
2147:
2145:
2142:
2139:
2138:
2116:
2113:
2112:
2111:
2071:
2068:
2054:
2052:
2026:
2001:
1955:
1939:
1936:
1902:69.147.212.194
1893:
1586:
1318:
965:
625:
612:
609:
608:
607:
606:
605:
604:
603:
576:
575:
574:
573:
485:
482:
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466:
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445:
432:
431:
418:
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412:
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389:
388:
377:
371:
370:
368:
351:the discussion
338:
337:
321:
309:
308:
300:
288:
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259:
245:
244:
235:
233:
232:
229:
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194:
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131:
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126:
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115:
106:
105:
103:
102:
95:
90:
81:
75:
73:
72:
61:
52:
51:
48:
47:
41:
25:
24:
19:
13:
10:
9:
6:
4:
3:
2:
3349:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3299:
3297:
3282:
3279:
3275:
3273:9780132683005
3270:
3266:
3265:Prentice Hall
3261:
3260:
3251:
3248:
3243:
3240:
3236:
3232:
3225:
3222:
3218:
3202:
3198:
3194:
3190:
3189:User:D.Lazard
3186:
3185:
3184:
3180:
3176:
3171:
3168:
3167:
3166:
3162:
3158:
3154:
3150:
3149:User:D.Lazard
3147:
3146:
3145:
3141:
3137:
3133:
3129:
3128:
3127:
3123:
3119:
3115:
3111:
3110:
3109:
3108:
3105:
3101:
3097:
3093:
3089:
3080:
3075:
3073:
3071:
3064:
3057:
3053:
3049:
3042:
3038:
3034:
3028:
3019:
3004:
2998:
2995:
2994:
2991:
2987:
2983:
2975:
2971:
2967:
2963:
2959:
2955:
2951:
2947:
2942:
2938:
2922:
2919:
2916:
2894:
2891:
2888:
2879:
2878:
2877:
2876:
2872:
2868:
2864:
2845:
2839:
2834:
2824:
2821:
2817:
2808:
2800:
2796:
2782:
2780:
2775:
2772:
2769:
2767:
2766:user:D.Lazard
2760:
2759:user:D.Lazard
2757:
2749:
2745:
2741:
2737:
2736:
2735:
2731:
2727:
2723:
2722:
2721:
2717:
2713:
2709:
2705:
2704:
2703:
2702:
2698:
2694:
2686:
2684:
2683:
2680:
2676:
2672:
2665:
2661:
2657:
2653:
2649:
2644:
2643:
2642:
2641:
2637:
2633:
2613:
2608:
2597:
2594:
2589:
2586:
2572:
2567:
2558:
2555:
2548:
2543:
2534:
2531:
2528:
2525:
2505:
2502:
2496:
2493:
2490:
2464:
2461:
2458:
2434:
2429:
2420:
2417:
2414:
2405:
2401:
2397:
2393:
2386:
2382:
2375:
2369:
2357:
2353:
2344:
2336:
2332:
2325:
2321:
2317:
2308:
2304:
2300:
2296:
2292:
2288:
2287:
2286:
2285:
2281:
2277:
2253:
2246:
2241:
2232:
2229:
2226:
2224:
2219:
2216:
2205:
2200:
2191:
2188:
2185:
2183:
2178:
2167:
2162:
2153:
2150:
2148:
2143:
2140:
2129:
2128:
2127:
2123:
2114:
2110:
2106:
2102:
2098:
2094:
2090:
2089:
2088:
2087:
2083:
2079:
2075:
2069:
2067:
2066:
2062:
2058:
2050:
2046:
2042:
2038:
2034:
2030:
2024:
2020:
2016:
2013:
2009:
2005:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1953:
1949:
1945:
1937:
1935:
1933:
1929:
1925:
1921:
1911:
1907:
1903:
1899:
1894:- Anonymous
1584:
1581:
1578:
1576:
1572:
1568:
1564:
1560:
1316:
1313:
963:
623:
620:
617:
610:
601:
597:
593:
589:
582:
581:
580:
579:
578:
577:
572:
568:
564:
560:
556:
552:
547:
546:
545:
541:
537:
533:
530:constitute a
529:
525:
521:
517:
513:
509:
508:
507:
505:
501:
497:
493:
483:
475:
470:
465:
464:
450:
449:
443:
439:
436:
435:
434:
433:
430:
426:
423:
420:
419:
415:
410:
405:
404:
401:
386:
382:
376:
373:
372:
369:
352:
348:
344:
343:
335:
329:
324:
322:
319:
315:
314:
310:
304:
301:
298:
294:
289:
285:
279:
271:
270:
260:
256:
251:
250:
231:
230:
225:
221:
213:
209:
205:
202:
200:
196:
195:
190:
186:
183:
180:
176:
172:
168:
165:
162:
159:
156:
153:
150:
147:
144:
140:
137:
136:Find sources:
133:
132:
124:
123:Verifiability
121:
119:
116:
114:
111:
110:
109:
100:
96:
94:
91:
89:
85:
82:
80:
77:
76:
70:
66:
65:Learn to edit
62:
59:
54:
53:
50:
49:
45:
39:
35:
31:
30:
23:
20:
18:
17:
3258:
3250:
3230:
3224:
3216:
3152:
3131:
3113:
3078:
3069:
3062:
3055:
3051:
3047:
3040:
3036:
3032:
3026:
3017:
2985:
2973:
2969:
2961:
2957:
2953:
2949:
2945:
2937:§ Congruence
2865:
2862:
2778:
2776:
2773:
2770:
2764:
2690:
2668:
2406:
2399:
2395:
2391:
2384:
2380:
2373:
2367:
2355:
2351:
2342:
2334:
2330:
2323:
2319:
2315:
2312:
2272:
2118:
2076:
2073:
2057:Jorge Stolfi
2048:
2044:
2040:
2036:
2032:
2028:
2022:
2018:
2014:
2011:
2007:
2003:
1997:
1993:
1992:by a number
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1951:
1947:
1943:
1941:
1924:94.73.55.201
1918:— Preceding
1914:
1896:— Preceding
1582:
1579:
1557:— Preceding
1554:
1314:
1311:
961:
621:
618:
614:
592:49.219.177.6
586:— Preceding
527:
523:
522:and nonzero
519:
515:
511:
490:— Preceding
487:
468:
413:
400:
381:Top-priority
380:
340:
306:Top‑priority
284:WikiProjects
267:
219:
197:
184:
178:
170:
163:
157:
151:
145:
135:
107:
32:This is the
2646:important.—
2407:Example 1.
1563:Jp.basuyaux
356:Mathematics
347:mathematics
303:Mathematics
161:free images
44:not a forum
3296:Categories
3217:References
3193:Nomen4Omen
3157:Nomen4Omen
3030:such that
2960:)), where
2907:1}" /: -->
2867:Nomen4Omen
2708:triple bar
3267:. p. 87.
3237:. p. 76.
3118:Anita5192
3092:zero ring
3018:congruent
2712:Anita5192
2648:Anita5192
2632:Karho.Yau
2389:, and so
2295:Anita5192
2276:Anita5192
2122:Vasily802
2097:Alpha3031
1555:Thanks.
1286:<<=
936:<<=
536:Anita5192
484:Division?
438:Archive 1
272:is rated
101:if needed
84:Be polite
34:talk page
3281:71081766
3242:77171950
3175:D.Lazard
3136:D.Lazard
3096:D.Lazard
3001:Given a
2982:D.Lazard
2884:1}": -->
2274:screen.—
1984:(modulo
1972:(modulo
1920:unsigned
1898:unsigned
1627:uint64_t
1615:uint64_t
1606:uint64_t
1597:uint64_t
1591:mul_mod2
1588:uint64_t
1571:contribs
1559:unsigned
1205:<<
1184:<<
1036:<<
864:<<
843:<<
666:uint64_t
654:uint64_t
645:uint64_t
636:uint64_t
627:uint64_t
588:unsigned
563:D.Lazard
492:unsigned
469:365 days
414:Archives
220:365 days
199:Archives
69:get help
42:This is
40:article.
3070:modulus
3020:modulo
2518:and so
2377:, then
2328:, then
630:mul_mod
619:But...
383:on the
274:C-class
167:WP refs
155:scholar
3153:moduli
3132:moduli
3081:< 2
2093:WP:BRD
1880:return
1541:return
1440:UInt64
1422:UInt64
1380:UInt64
1365:UInt64
1353:UInt32
1344:UInt64
1335:UInt64
1329:ModPow
1326:UInt64
1323:static
1320:public
1298:return
1063:UInt64
1048:UInt64
1027:UInt64
1015:UInt64
1000:UInt64
991:UInt64
982:UInt64
976:ModPow
973:UInt64
970:static
967:public
948:return
280:scale.
139:Google
3065:: -->
3060:. If
3054:(mod
3039:(mod
2952:(mod
2941:WP:OR
2909:into
2892:: -->
2398:(mod
2387:) = 1
2337:(mod
2322:(mod
2043:with
1867:: -->
1866:: -->
1728:&
1704:while
1682:: -->
1651:: -->
1456:: -->
1455:: -->
1434:&
1401:while
1262:: -->
1223:&
1166:: -->
1076:: -->
1075:: -->
1012:const
912:: -->
882:&
825:: -->
749:: -->
718:: -->
691:: -->
690:: -->
532:field
422:Index
261:This
204:Index
182:JSTOR
143:books
97:Seek
3278:LCCN
3269:ISBN
3239:LCCN
3197:talk
3179:talk
3161:talk
3140:talk
3122:talk
3100:talk
3012:and
2976:= 1.
2871:talk
2744:talk
2730:talk
2716:talk
2697:talk
2679:WT79
2652:talk
2636:talk
2451:and
2379:gcd(
2350:gcd(
2299:talk
2280:talk
2082:talk
2061:talk
1948:real
1928:talk
1906:talk
1842:else
1812:<
1779:else
1749:<
1567:talk
1226:Mask
1133:<
1018:Mask
792:<
596:talk
567:talk
540:talk
534:. —
500:talk
175:FENS
149:news
86:and
2964:is
2609:mod
2568:mod
2544:mod
2485:gcd
2430:mod
2320:k b
2316:k a
2242:mod
2201:mod
2163:mod
2025:⊆ .
1996:in
1946:of
1883:sum
1782:sum
1767:sum
1746:sum
1630:sum
1472:Bit
1425:Bit
1169:MP2
1115:int
1109:for
1066:MP2
828:mp2
771:for
700:int
681:mp2
553:or
375:Top
189:TWL
3298::
3276:.
3199:)
3181:)
3163:)
3142:)
3124:)
3102:)
3050:≡
3035:≡
2992:)
2984::
2948:≡
2920:≥
2901:1}
2873:)
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