226:, maybe a discussion about this will be interesting. The wording "... is irreducible if the coefficients 1 and −2 are considered as integers" has, in my interpretation, two problems. The first is the obvious one: that it appears to try to draw a distinction between the integers as elements of the ring of integers, and the integers as elements of a subring of the reals. This is like claiming that the set of integers cannot be regarded as a subset of the reals (or, to belabour the point, that integers cannot be
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polynomials with integer coefficients", which is what is really meant. The person who makes no such implicit inferences might simply interpret it at face value: that the coefficients of the initial polynomial are integers. Face it, "considered as" is not a formally defined phrase in mathematics (and has even less meaning to the newbie); it is a cue to look for unstated inferences. What about "... is irreducible if it is to be factored as polynomials with
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product of two non-constant polynomials" being a definition cannot be true or false. The precise definition, when the coefficients do not belong to a field, is too technical and not enough commonly used for belonging to the lead. This a reason for having "roughly" in the first sentence. The other reason for this "roughly" is the ambiguity of "cannot be factored", explained in the next sentences.
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claims the above. This is true for the second definition of irreducibility, where reducibles factor into two non-constant polynomials, but unless I'm mistaken about something, it seems false for the first definition. Counter example: 3x+3 is reducible over the integers because it equals 3(x+1), and 3
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I agree that "prime" and "irreducible" are not always equivalent. Moreover, "prime polynomial" is rarely used, except when proving the equivalence over a field. Thus, I have edited the parenthesis accordingly. On the other hand the statement "a non-constant polynomial that cannot be factored into the
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are supposed to belong". I'm not convinced that absolute irreducibility is relevant to the discussion at all. As I understand it, all that is being said is that an irreducible polynomial has exactly one "nontrivial" (more technically, "prime") factor in the polynomial ring under consideration. The
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It rambles around the idea that roots might have "no explicit algebraic expression". This is not what Abel-Ruffini
Theorem says. Note that $ \sqrt{2}$ is as algebraic as the root of every other polynomial. The section has no meaning unless it states correctly the type of expression that is claiming
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The introduction seems to make strong assumptions about the ring the polynomial is defined over. In particular, an irreducible polynomial is not necessarily the same thing as a prime element over a general integral domain, and the statement "a non-constant polynomial that cannot be factored into the
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I'm not sure the concern is properly addressed. One should not assume that the reader makes the "standard" assumptions that someone familiar with the topic does. The person familiar topic might read "considered as a polynomial with integer coefficients" to mean "all operations being in the ring of
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As, over an integral domain, the total degree of a product of polynomials is the sum of the total degrees of the factors, the units of a polynomial ring are exactly the same as the units of the integral domain of coefficients. So, both formulations are equivalent. The one of the article is simpler,
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I'll change "if the coefficients 1 and −2 are considered as integers" into "if the polynomial is considered as a polynomial with integer coefficients". The formulation "if the polynomial is considered as a polynomial over the integers" is also fine for me, but may be considered as too technical or
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being real numbers). The second is more subtle, and is my primary objection: the factorization constraint specifically applies to the coefficients of the factor polynomials (by definition, also to the polynomial to be factored, but that does not make a difference here). The wording as it stands
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I think it's worth talking about rewriting the lede - someone I was helping a little while ago was also thrown off by this and it looks like it's caused confusion at various times. I would prefer talking about the existence of a "non-trivial factorisation" and then clarifying what this means for
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It is questionable whether the clarification offered by the section is needed. The definition(s) state where the coefficients are taken from and if the coefficients are the complex numbers, then the definition allows all complex numbers in the factorization, potentially even using non-algebraic
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But it's a non-invertible and non-zero polynomial over the integers, right? If that is correct, then it's also non-constant according to this sentence: "...in this case, the non-constant polynomials are exactly the polynomials that are non-invertible and non-zero." I agree that
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above. Ultimately the problem stems from trying to write an article that covers the cases of coefficients in a field and in other rings simultaneously. It does not seem like an easy problem to solve, but any attempt should probably begin by reading those talk-page sections.
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states that constraint applies specifically and only to the specific coefficients of the polynomial to be factored, which would only be useful if the first problem did not exist. Do you at least see where I'm coming from? —
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Both definitions generalize the definition given for the case of coefficients in a field, because, in this case, the non-constant polynomials are exactly the polynomials that are non-invertible and non-zero.
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definition section is pretty clear on this, and has reasonably concise and understandable language. However, my primary concern of possibly incorrect initial interpretation has at least been addressed. —
2204:: if -1 or 2 is a quadratic residue, then the first or the third factorization holds. If none is a quadratic residue, then their product –2 is a quadratic residue, and the second factorization holds.
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The last constraint seems to exclude the factor polynomials that have even one unit coefficient, but this can't be right. I think the intention might have been "... and which are not
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product of two non-constant polynomials" is obviously only true for fields. While the definition section clarifies this, it seems unnecessarily misleading to the casual reader.
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I have edited the lead in an attempt to clarify this. Feel free to improve my wording. This discussion is, in fact, about the distinction between irreducibility and
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In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials.
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I've tweaked the main thing that was bothering me. The secondary one (that you have responded to) may not be worth addressing as being natural enough. —
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integral domains and fields. Changes have been talked about above but I still think it's all unclear, at the very least it should be pointed out that
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is never considered as non-constant, and I do not see any sentence of the article that could implies that it could be considered as non-constant.
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might or might not exist and which existence or not does not affect irreducibility. It seems that the intent of the section was to refer to
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In fact, the concern is not really about the sentence that has been discussed, but about the preceding one. I'll try to address this.
907:: We should state that it is irreducible according to one definition, but reducible according to another, and (briefly) explain why.
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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My primary concern is essentially addressed by the clarification "the field or ring to which the coefficients of the polynomial
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The referent of "this case" in that sentence is "the case when the coefficients come from a field". That could be clearer. --
1965:, which is possible if -2 is a quadratic residue modulo p, that is p = 1 or 3 mod 8. The case p = 5 mod 8 is not resolved. Is
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is an ID not a field and so we are looking at the other criterion. Wanted to discuss here before making any major changes. --
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self-reply Answer is YES. If p = 5 mod 8, (implying p = 1 mod 4) then -1 is a quadratic residue. Let's call r = √-1.
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Yeah, I thought "this case" referred to the case described in the previous sentence. Thanks for clearing this up!
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numbers. Anyway, if the section is to exist, it should at least have the correct statement of what is claiming.
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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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section more closely. Note also that it is not really sensible to treat a constant polynomial an element of
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Good point. I have edited the article for clarifying that the second definition is not used in this article.
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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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polynomial over the integers. I find it hard to reconcile this with any reasonable definition of
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Is this analysis correct? If yes, we should probably refine the claims about irreducibility of
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is irreducible according to one definition, but reducible according to another definition.
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A polynomial with integer coefficients, or, more generally, with coefficients in a
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is not a unit. However, it is not reducible over F_5, because 3 is a unit there.
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https://archive.is/20130101095630/http://theory.cs.uvic.ca/inf/neck/PolyInfo.html
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is irreducible over the rationals, so according to this definition it is also
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if it is not the product of two polynomials that have their coefficients in
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should not be considered non-constant. I'd say that sentence is misleading.
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would be considered a "non-constant" polynomial. What do you think?
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Another definition is frequently used, saying that a polynomial is
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The section Nature of a factor has ambiguous/incorrect language
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Your first post follows perfectly the recommendations of the
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as may have been intended, despite the natural embedding of
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in the polynomial ring", since this would have fitted the
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I have added a paragraph in the lead for clarifying this.
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for additional information. I made the following changes:
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This seems contradictory, as the first quote implies that
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but if you find it confusing, be free of changing it.
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This was ambiguous, and is (I guess) now clarified.
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2078:{\displaystyle \mathbb {F} _{29}}
2049:{\displaystyle \mathbb {F} _{13}}
1059:Irreducible polynomial#Definition
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1522:
1475:
1407:
1406:
1402:
1377:manual of style
1353:
1248:
1247:
1238:
1191:
1190:
1143:
1142:
1134:
1071:
1070:
1055:
1040:
1035:
1003:
996:have permission
986:
960:this simple FaQ
945:
878:
873:
872:
835:
830:
829:
759:
754:
753:
734:
733:
703:
689:
688:
658:
653:
652:
637:polynomial ring
575:
570:
569:
511:
421:
391:
366:
361:
360:
330:
221:
207:
196:
190:
172:
122:
119:
116:
113:
112:
90:
83:
63:
34:on Knowledge's
31:
12:
11:
5:
2247:
2245:
2237:
2236:
2231:
2221:
2220:
2217:
2216:
2197:
2196:
2171:
2167:
2163:
2158:
2154:
2150:
2146:
2142:
2139:
2134:
2130:
2125:
2120:
2116:
2113:
2108:
2104:
2099:
2072:
2067:
2043:
2038:
2014:
2009:
1987:
1984:
1979:
1975:
1953:
1949:
1946:
1943:
1938:
1935:
1930:
1925:
1921:
1916:
1911:
1907:
1904:
1901:
1896:
1893:
1888:
1883:
1879:
1874:
1851:
1846:
1823:
1819:
1816:
1813:
1808:
1803:
1798:
1794:
1789:
1784:
1780:
1777:
1774:
1769:
1764:
1759:
1755:
1750:
1737:. Is that so?
1724:
1719:
1696:
1675:
1672:
1667:
1663:
1650:
1647:
1646:
1645:
1644:
1643:
1631:
1630:
1589:
1588:
1587:
1586:
1577:, and are not
1569:is said to be
1557:this statement
1549:
1548:Error in lead?
1546:
1521:
1518:
1517:
1516:
1474:
1471:
1470:
1469:
1415:
1401:
1398:
1397:
1396:
1381:
1352:
1349:
1348:
1347:
1346:
1345:
1344:
1343:
1342:
1341:
1340:
1339:
1338:
1337:
1336:
1335:
1270:
1267:
1264:
1261:
1258:
1255:
1213:
1210:
1207:
1204:
1201:
1198:
1165:
1162:
1159:
1156:
1153:
1150:
1093:
1090:
1087:
1084:
1081:
1078:
1054:
1053:"Non-constant"
1051:
1030:
1029:
1022:
975:
974:
966:Added archive
944:
941:
940:
939:
938:
937:
936:
935:
934:
933:
896:
893:
890:
885:
881:
869:
853:
850:
847:
842:
838:
827:
816:(the field of
800:
793:
777:
774:
771:
766:
762:
741:
721:
718:
715:
710:
706:
702:
699:
696:
676:
673:
670:
665:
661:
650:
631:) if it is an
612:
605:
593:
590:
587:
582:
578:
561:
560:
559:
558:
509:
503:
502:
491:
488:
485:
480:
475:
472:
469:
466:
463:
458:
453:
450:
447:
444:
439:
436:
433:
428:
424:
420:
417:
414:
411:
406:
403:
398:
394:
390:
387:
384:
381:
378:
373:
369:
354:
353:
340:
339:
329:
326:
325:
324:
323:
322:
321:
320:
319:
318:
282:
220:
217:
214:
213:
201:
198:
197:
192:
188:
186:
183:
182:
174:
173:
168:
162:
155:
154:
151:
150:
147:
146:
135:
129:
128:
126:
109:the discussion
96:
95:
79:
67:
66:
58:
46:
45:
39:
28:
13:
10:
9:
6:
4:
3:
2:
2246:
2235:
2232:
2230:
2227:
2226:
2224:
2215:
2211:
2207:
2199:
2198:
2195:
2191:
2187:
2169:
2165:
2161:
2156:
2152:
2148:
2144:
2140:
2137:
2132:
2128:
2123:
2118:
2114:
2111:
2106:
2102:
2097:
2088:
2087:
2086:
2070:
2041:
2012:
1985:
1982:
1977:
1973:
1951:
1947:
1944:
1941:
1936:
1933:
1928:
1923:
1919:
1914:
1909:
1905:
1902:
1899:
1894:
1891:
1886:
1881:
1877:
1872:
1849:
1821:
1817:
1814:
1811:
1806:
1801:
1796:
1792:
1787:
1782:
1778:
1775:
1772:
1767:
1762:
1757:
1753:
1748:
1738:
1722:
1673:
1670:
1665:
1661:
1648:
1642:
1639:
1635:
1634:
1633:
1632:
1629:
1625:
1621:
1616:
1615:
1614:
1613:
1610:
1598:
1594:
1580:
1572:
1564:
1563:
1562:
1561:
1560:
1558:
1554:
1547:
1545:
1544:
1540:
1536:
1530:
1528:
1519:
1515:
1511:
1507:
1504:
1500:
1496:
1495:
1494:
1493:
1489:
1485:
1480:
1472:
1468:
1464:
1460:
1456:
1455:
1454:
1453:
1448:
1444:
1440:
1436:
1430:
1399:
1395:
1391:
1387:
1382:
1378:
1373:
1372:
1371:
1370:
1366:
1362:
1357:
1350:
1334:
1330:
1326:
1322:
1321:
1320:
1316:
1312:
1308:
1307:
1306:
1302:
1298:
1294:
1293:
1292:
1288:
1284:
1268:
1265:
1259:
1253:
1242:
1237:
1236:
1235:
1231:
1227:
1211:
1208:
1202:
1196:
1189:
1188:
1187:
1183:
1179:
1163:
1160:
1154:
1148:
1138:
1133:
1132:
1130:
1129:
1128:
1127:
1123:
1119:
1115:
1111:
1107:
1091:
1088:
1082:
1076:
1066:
1062:
1060:
1052:
1050:
1049:
1044:
1039:
1038:
1027:
1023:
1020:
1016:
1015:
1014:
1007:
1001:
997:
993:
989:
983:
978:
973:
969:
965:
964:
963:
961:
957:
953:
948:
942:
932:
928:
924:
920:
919:
918:
914:
910:
891:
883:
879:
870:
867:
848:
840:
836:
828:
825:
823:
819:
815:
811:
807:
801:
798:
794:
791:
772:
764:
760:
739:
716:
713:
708:
704:
700:
694:
671:
663:
659:
651:
648:
646:
642:
638:
634:
630:
626:
622:
619:
613:
610:
606:
588:
580:
576:
567:
566:
565:
564:
563:
562:
557:
553:
549:
544:
540:
536:
532:
528:
527:
526:
525:
524:
523:
519:
515:
508:
486:
483:
478:
473:
464:
461:
456:
451:
445:
442:
434:
431:
426:
422:
418:
412:
409:
404:
401:
396:
392:
388:
385:
379:
371:
367:
359:
358:
357:
352:
350:
345:
344:
343:
338:
335:
334:
333:
328:Contradiction
327:
317:
314:
309:
305:
304:
303:
299:
295:
291:
287:
283:
281:
277:
273:
269:
268:
267:
264:
260:
255:
254:
253:
249:
245:
240:
239:
238:
237:
234:
229:
228:considered as
225:
224:Joel B. Lewis
218:
210:
205:
200:
199:
185:
184:
181:
180:
176:
175:
171:
166:
161:
160:
144:
140:
134:
131:
130:
127:
110:
106:
102:
101:
93:
87:
82:
80:
77:
73:
72:
68:
62:
59:
56:
52:
47:
43:
37:
29:
25:
20:
19:
16:
2184:. Et voilà.
1739:
1652:
1596:
1590:
1570:
1551:
1531:
1523:
1502:
1477:The section
1476:
1403:
1358:
1354:
1114:non-constant
1113:
1109:
1106:non-constant
1105:
1068:
1064:
1056:
1034:
1031:
1006:source check
985:
979:
976:
949:
946:
865:
821:
813:
805:
803:
789:
732:, and since
644:
628:
624:
620:
615:
506:
504:
355:
346:
341:
336:
331:
307:
227:
222:
203:
177:
169:
139:Mid-priority
138:
98:
64:Mid‑priority
42:WikiProjects
15:
1571:irreducible
1443:CentralAuth
1309:Clarified.
866:irreducible
625:irreducible
114:Mathematics
105:mathematics
61:Mathematics
32:Start-class
2223:Categories
1597:Definition
1043:Report bug
641:invertible
539:Definition
1026:this tool
1019:this tool
790:reducible
347:Over the
2206:D.Lazard
1620:D.Lazard
1553:D.Lazard
1506:D.Lazard
1459:D.Lazard
1439:Contribs
1429:Caliburn
1386:D.Lazard
1325:Chrisahn
1311:D.Lazard
1283:Chrisahn
1241:D.Lazard
1226:D.Lazard
1178:Chrisahn
1137:D.Lazard
1118:Chrisahn
1110:constant
1032:Cheers.—
923:D.Lazard
909:Chrisahn
349:integers
294:D.Lazard
272:D.Lazard
244:D.Lazard
242:jargon.
204:365 days
170:Archives
1638:Quondum
1609:Quondum
956:my edit
635:of the
313:Quondum
263:Quondum
259:integer
233:Quondum
141:on the
541:, and
514:Loraof
38:scale.
1503:Fixed
1057:From
820:, if
2210:talk
2190:talk
1624:talk
1593:unit
1579:unit
1539:talk
1510:talk
1488:talk
1463:talk
1435:Talk
1390:talk
1365:talk
1329:talk
1315:talk
1301:talk
1287:talk
1230:talk
1182:talk
1122:talk
927:talk
913:talk
627:(or
552:talk
518:talk
298:talk
276:talk
248:talk
2085:…?
1581:in
1447:Log
1297:JBL
1000:RfC
970:to
812:of
788:is
548:JBL
133:Mid
2225::
2212:)
2192:)
2162:−
2112:−
2071:29
2056:,
2042:13
2027:,
1945:−
1934:−
1929:−
1903:−
1892:−
1802:−
1626:)
1585:."
1541:)
1529:.
1512:)
1490:)
1465:)
1445:·
1441:·
1437:·
1431:·
1392:)
1367:)
1331:)
1317:)
1303:)
1289:)
1232:)
1184:)
1124:)
1112:/
1013:.
1008:}}
1004:{{
929:)
915:)
714:−
554:)
546:--
537:,
533:,
520:)
462:−
432:−
402:−
300:)
278:)
250:)
2208:(
2202:p
2188:(
2170:2
2166:r
2157:4
2153:X
2149:=
2145:)
2141:r
2138:+
2133:2
2129:X
2124:(
2119:)
2115:r
2107:2
2103:X
2098:(
2066:F
2037:F
2013:5
2008:F
1986:1
1983:+
1978:4
1974:X
1952:)
1948:1
1942:x
1937:2
1924:2
1920:x
1915:(
1910:)
1906:1
1900:x
1895:2
1887:+
1882:2
1878:x
1873:(
1850:p
1845:F
1822:)
1818:1
1815:+
1812:x
1807:2
1797:2
1793:x
1788:(
1783:)
1779:1
1776:+
1773:x
1768:2
1763:+
1758:2
1754:x
1749:(
1723:b
1718:F
1695:Z
1674:1
1671:+
1666:4
1662:X
1622:(
1605:R
1601:R
1583:R
1575:R
1567:R
1537:(
1508:(
1486:(
1461:(
1449:)
1433:(
1414:Z
1388:(
1379:.
1363:(
1327:(
1313:(
1299:(
1285:(
1269:3
1266:=
1263:)
1260:x
1257:(
1254:p
1243::
1239:@
1228:(
1212:3
1209:=
1206:)
1203:x
1200:(
1197:p
1180:(
1164:3
1161:=
1158:)
1155:x
1152:(
1149:p
1139::
1135:@
1120:(
1092:3
1089:=
1086:)
1083:x
1080:(
1077:p
1045:)
1041:(
1028:.
1021:.
925:(
911:(
895:)
892:x
889:(
884:3
880:p
852:)
849:x
846:(
841:3
837:p
822:R
814:R
799::
776:)
773:x
770:(
765:3
761:p
740:3
720:)
717:1
709:2
705:x
701:3
698:(
695:3
675:)
672:x
669:(
664:3
660:p
647:.
645:R
621:R
611::
592:)
589:x
586:(
581:3
577:p
550:(
516:(
510:3
507:p
490:)
487:1
484:+
479:3
474:x
471:(
468:)
465:1
457:3
452:x
449:(
446:3
443:=
438:)
435:1
427:2
423:x
419:3
416:(
413:3
410:=
405:3
397:2
393:x
389:9
386:=
383:)
380:x
377:(
372:3
368:p
296:(
274:(
246:(
179:1
145:.
44::
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