2177:
has been translated from Latin into
English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
491:
The logic of this proof is basically Euclid's, but the notation and some of the concepts (zero, negative) would be foreign to him. It relies on the fact that a set of non-negative integers has a smallest member.
458:
401:
333:
1408:
is finite, therefore their intersection is finite and has a largest member. It must be positive because 1 is a positive common divisor of
76:
The lemma is not true for composite numbers. For example, 8 does not divide 4 and 8 does not divide 6, yet 8 does divide their product 24.
2283:
2338:
2307:
2236:
70:
2299:
2188:
2196:
21:
2229:
Untersuchungen uber hohere
Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
2173:
61:. It states that if a prime divides the product of two numbers, it must divide one of the factors. For example since
2275:
1315:
271:
17:
1055:
35:
253:
2095:
54:
31:
416:
2334:
2303:
2279:
2245:
2232:
2205:
2192:
368:
300:
206:
2257:
2217:
58:
42:
89:
257:. It is included in practically every book that covers elementary number theory.
2183:
Gauss, Carl
Friedrich; Clarke, Arthur A. (translator into English) (1986),
65:
is divisible by 19, one or both of 133 or 143 must be as well (In fact,
2331:
Prime
Numbers and Computer Methods for Factorization (second edition)
1287:
is non-negative and is less than the least positive common multiple
2227:
Gauss, Carl
Friedrich; Maser, H. (translator into German) (1965),
971:− 1 the corresponding remainder would be greater than or equal to
2296:
A Classical
Introduction to Modern Number Theory (Second edition)
483:. This is the generalization of Euclid's lemma mentioned above.
270:
The easiest proof of Euclid's lemma uses another lemma called
913:
an integer} contains both positive and negative numbers. Let
1382:≠ 0, and their common divisors are simply the divisors of
1118:| so it is not empty and therefore has a smallest member.
1372:≠ 0 they have a greatest common divisor. It is positive.
251:
The lemma first appears as proposition 30 in Book VII of
2272:
2185:
1129:≠ 0 their least positive common multiple is written lcm(
1114:
Proof. The set of positive common multiples contains |
851:
is an integer then there is a unique pair of integers
925:
be the smallest non-negative number in the set. Then
419:
371:
303:
284:are relatively prime integers there exist integers
57:that captures one of the fundamental properties of
452:
395:
327:
1867:is defined to be the greatest common divisor, so
1427:≠ 0 their greatest common divisor is written gcd(
1112:≠ 0 they have a least positive common multiple.
190:A generalization is also called Euclid's lemma:
8:
463:The first term on the left is divisible by
2294:Ireland, Kenneth; Rosen, Michael (1990),
418:
370:
302:
2107:
901:Proof. (existence) The set of numbers {
508:be any integer. If there is an integer
2253:
2243:
2213:
2203:
835:only has a finite number of divisors.
467:, and the second term is divisible by
223:(This is a generalization because if
346:be relatively prime, and assume that
7:
2270:Hardy, G. H.; Wright, E. M. (1980),
471:which by hypotheses is divisible by
96:divides the product of two integers
1374:Proof. Every integer divides 0. If
1404:is finite, the set of divisors of
955:were replaced by a smaller value,
28:
1579:. The proof has two steps i) if
1015:the remainder would be negative:
71:fundamental theorem of arithmetic
2132:Ireland & Rosen, prop. 1.1.1
104:. (In symbols this is written
1:
1745:ii) Since any common divisor
1887:Relation between gcd and lcm
1400:≠ 0. The set of divisors of
69:It used in the proof of the
2174:Disquisitiones Arithmeticae
1832:is an integer. Similarly,
1663:It is a common multiple of
1567:, i.e. there is an integer
1386:. The largest of these is |
2373:
2123:Hardy & Wright, Thm. 3
1462:be any common divisor of
1164:be any common multiple of
453:{\displaystyle rnb+sab=b.}
133:Equivalent statements are
29:
1275:is a commmon multiple of
1183:Proof. There are numbers
839:Division with a remainder
2320:Elementary Number Theory
2054:is a common multiple of
1551:is a common multiple of
396:{\displaystyle rn+sa=1.}
328:{\displaystyle rx+sy=1.}
265:
30:Not to be confused with
2318:Landau, Edmund (1966),
2276:Oxford University Press
1855:is a common divisor of
1613:be a common divisor of
1316:Greatest common divisor
479:, is also divisible by
475:. Therefore their sum,
406:Multiply both sides by
354:. By Bézout, there are
274:. This states that if
239:is relatively prime to
2333:, Boston: Birkhäuser,
2062:, it is a multiple of
2004:, 0) = 1 implies that
454:
397:
329:
51:Euclid's first theorem
18:User:Virginia-American
2329:Riesel, Hans (1994),
2231:, New York: Chelsea,
1952:
1648:Consider the number
1621:. There are integers
1583:is common divisor of
1527:Now assume that both
1517:trivially imply that
1392:Now assume that both
1056:Least common multiple
831:|. This implies that
455:
398:
330:
266:Via Bézout's identity
417:
369:
301:
2322:, New York: Chelsea
2096:Euclidean algorithm
1211:. It follows from
733:is an integer then
500:Definition. Assume
130:(or perhaps both).
2256:has generic name (
2216:has generic name (
2114:Gauss, DA, Art. 14
1489:≠ 0. In this case
797:are integers then
450:
393:
325:
36:Euclid's algorithm
2150:Riesel, Thm. A2.1
1772:In particular,
1683:. Multiplying by
272:Bézout's identity
254:Euclid's Elements
227:is prime, either
63:133 × 143 = 19019
2364:
2343:
2323:
2312:
2288:
2261:
2255:
2251:
2249:
2241:
2221:
2215:
2211:
2209:
2201:
2160:
2157:
2151:
2148:
2142:
2139:
2133:
2130:
2124:
2121:
2115:
2112:
2008:= ±1 . But then
1850:
1823:
1800:
1790:
1771:
1743:
1721:and dividing by
1720:
1701:
1671:, so there is a
1662:
1647:
1438:Theorem. Assume
1140:Theorem. Assume
951:(uniqueness) If
459:
457:
456:
451:
402:
400:
399:
394:
334:
332:
331:
326:
293:
283:
207:relatively prime
166:does not divide
156:does not divide
148:does not divide
140:does not divide
68:
64:
32:Euclid's theorem
2372:
2371:
2367:
2366:
2365:
2363:
2362:
2361:
2359:
2356:
2353:
2350:
2347:
2341:
2328:
2317:
2310:
2293:
2286:
2269:
2265:
2252:
2242:
2239:
2226:
2212:
2202:
2199:
2182:
2169:
2164:
2163:
2158:
2154:
2149:
2145:
2141:Landau, Thm. 15
2140:
2136:
2131:
2127:
2122:
2118:
2113:
2109:
2104:
2092:
2017:
1955:
1937:) = 1 then lcm(
1920:
1889:
1833:
1806:
1792:
1773:
1762:
1744:
1726:
1707:
1702:Multiplying by
1688:
1649:
1630:
1608:
1526:
1391:
1373:
1341:is any integer
1321:Definition. If
1319:
1113:
1081:is any integer
1071:common multiple
1061:Definition. If
1059:
950:
841:
810:
772:
742:
720:
694:
672:
654:
612:
602:
498:
489:
487:Euclidean proof
415:
414:
367:
366:
299:
298:
285:
275:
268:
263:
249:
222:
189:
161:
113:
82:
66:
62:
39:
26:
25:
24:
12:
11:
5:
2370:
2368:
2345:
2344:
2339:
2325:
2324:
2314:
2313:
2308:
2290:
2289:
2285:978-0198531715
2284:
2263:
2262:
2237:
2223:
2222:
2197:
2168:
2165:
2162:
2161:
2152:
2143:
2134:
2125:
2116:
2106:
2105:
2103:
2100:
2099:
2098:
2091:
2088:
1954:
1953:Euclid's lemma
1951:
1921:Corollary. If
1891:Corollary. If
1888:
1885:
1875:|. Therefore,
1331:common divisor
1318:
1313:
1295:= 0. But then
1058:
1053:
1007:Similarly, if
840:
837:
497:
494:
488:
485:
461:
460:
449:
446:
443:
440:
437:
434:
431:
428:
425:
422:
404:
403:
392:
389:
386:
383:
380:
377:
374:
336:
335:
324:
321:
318:
315:
312:
309:
306:
267:
264:
262:
259:
248:
245:
81:
78:
67:19 × 7 = 133.)
47:Euclid's lemma
27:
15:
14:
13:
10:
9:
6:
4:
3:
2:
2369:
2360:
2357:
2354:
2351:
2348:
2342:
2340:0-8176-3743-5
2336:
2332:
2327:
2326:
2321:
2316:
2315:
2311:
2309:0-387-97329-X
2305:
2301:
2297:
2292:
2291:
2287:
2281:
2277:
2273:
2268:
2267:
2266:
2259:
2254:|first2=
2247:
2240:
2238:0-8284-0191-8
2234:
2230:
2225:
2224:
2219:
2214:|first2=
2207:
2200:
2194:
2190:
2186:
2181:
2180:
2179:
2176:
2175:
2166:
2159:Landau, ch. 1
2156:
2153:
2147:
2144:
2138:
2135:
2129:
2126:
2120:
2117:
2111:
2108:
2101:
2097:
2094:
2093:
2089:
2087:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2034:). Since gcd(
2033:
2029:
2025:
2021:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1987:
1984:Proof. Since
1982:
1980:
1976:
1972:
1968:
1964:
1960:
1950:
1948:
1944:
1940:
1936:
1932:
1928:
1924:
1918:
1914:
1910:
1906:
1902:
1899:≠ 0 then lcm(
1898:
1894:
1886:
1884:
1882:
1878:
1874:
1870:
1866:
1862:
1858:
1854:
1848:
1844:
1840:
1836:
1831:
1827:
1821:
1817:
1813:
1809:
1804:
1799:
1795:
1788:
1784:
1780:
1776:
1769:
1765:
1760:
1756:
1752:
1748:
1741:
1737:
1733:
1729:
1724:
1718:
1714:
1710:
1705:
1699:
1695:
1691:
1686:
1682:
1678:
1674:
1670:
1666:
1660:
1656:
1652:
1645:
1641:
1637:
1633:
1628:
1624:
1620:
1616:
1612:
1606:
1602:
1598:
1594:
1590:
1586:
1582:
1578:
1574:
1570:
1566:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1535:≠ 0, and let
1534:
1530:
1524:
1520:
1516:
1512:
1508:
1504:
1500:
1496:
1492:
1488:
1484:
1479:
1477:
1473:
1469:
1465:
1461:
1457:
1453:
1449:
1445:
1441:
1436:
1434:
1430:
1426:
1422:
1419:Notation. If
1417:
1415:
1411:
1407:
1403:
1399:
1395:
1389:
1385:
1381:
1377:
1371:
1367:
1362:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1317:
1314:
1312:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1181:
1179:
1175:
1171:
1167:
1163:
1159:
1155:
1151:
1147:
1143:
1138:
1136:
1132:
1128:
1124:
1121:Notation. If
1119:
1117:
1111:
1107:
1102:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1057:
1054:
1052:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
978:
974:
970:
966:
962:
958:
954:
948:
944:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
899:
897:
893:
889:
885:
880:
878:
874:
870:
866:
862:
858:
854:
850:
846:
838:
836:
834:
830:
826:
822:
818:
814:
808:
804:
800:
796:
792:
788:
784:
780:
776:
770:
766:
762:
758:
754:
750:
746:
740:
736:
732:
728:
724:
718:
714:
710:
706:
702:
698:
692:
688:
684:
680:
676:
670:
666:
662:
658:
652:
648:
644:
640:
636:
632:
628:
624:
620:
616:
610:
606:
600:
596:
592:
588:
584:
580:
575:
573:
569:
565:
561:
556:
554:
550:
546:
542:
538:
534:
530:
527:
523:
519:
515:
511:
507:
503:
495:
493:
486:
484:
482:
478:
474:
470:
466:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
413:
412:
411:
409:
390:
387:
384:
381:
378:
375:
372:
365:
364:
363:
361:
357:
353:
349:
345:
341:
322:
319:
316:
313:
310:
307:
304:
297:
296:
295:
292:
288:
282:
278:
273:
260:
258:
256:
255:
246:
244:
242:
238:
234:
230:
226:
220:
216:
212:
208:
204:
200:
196:
191:
187:
185:
181:
177:
173:
169:
165:
159:
155:
151:
147:
143:
139:
134:
131:
129:
125:
121:
117:
111:
107:
103:
99:
95:
92:, and assume
91:
87:
79:
77:
74:
72:
60:
59:prime numbers
56:
52:
49:(also called
48:
44:
43:number theory
37:
33:
23:
19:
2358:
2355:
2352:
2349:
2346:
2330:
2319:
2298:, New York:
2295:
2271:
2264:
2228:
2187:, New York:
2184:
2172:
2170:
2155:
2146:
2137:
2128:
2119:
2110:
2083:
2079:
2075:
2071:
2067:
2063:
2059:
2055:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
2019:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1983:
1978:
1974:
1970:
1966:
1962:
1958:
1957:Theorem. If
1956:
1946:
1942:
1938:
1934:
1930:
1929:≠ 0 and gcd(
1926:
1922:
1916:
1912:
1908:
1904:
1900:
1896:
1892:
1890:
1880:
1876:
1872:
1868:
1864:
1860:
1856:
1852:
1846:
1842:
1838:
1834:
1829:
1825:
1819:
1815:
1811:
1807:
1802:
1797:
1793:
1786:
1782:
1778:
1774:
1767:
1763:
1758:
1754:
1750:
1746:
1739:
1735:
1731:
1727:
1722:
1716:
1712:
1708:
1703:
1697:
1693:
1689:
1684:
1680:
1676:
1672:
1668:
1664:
1658:
1654:
1650:
1643:
1639:
1635:
1631:
1626:
1622:
1618:
1614:
1610:
1604:
1600:
1596:
1592:
1588:
1584:
1580:
1576:
1572:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1528:
1522:
1518:
1514:
1510:
1506:
1502:
1498:
1494:
1490:
1486:
1482:
1480:
1475:
1471:
1467:
1463:
1459:
1455:
1451:
1447:
1443:
1439:
1437:
1432:
1428:
1424:
1420:
1418:
1413:
1409:
1405:
1401:
1397:
1393:
1387:
1383:
1379:
1375:
1369:
1365:
1363:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1322:
1320:
1308:
1304:
1300:
1296:
1292:
1291:. Therefore
1288:
1284:
1280:
1276:
1272:
1268:
1264:
1260:
1256:
1252:
1248:
1247:. Similarly
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1208:
1204:
1200:
1196:
1192:
1188:
1184:
1182:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1149:
1145:
1141:
1139:
1134:
1130:
1126:
1122:
1120:
1115:
1109:
1105:
1103:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1060:
1048:
1044:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1004:
1000:
996:
992:
988:
984:
980:
976:
972:
968:
964:
960:
956:
952:
946:
942:
938:
934:
930:
926:
922:
918:
914:
910:
906:
902:
900:
895:
891:
887:
883:
882:Definition.
881:
876:
872:
868:
864:
860:
856:
852:
848:
844:
843:Theorem. If
842:
832:
828:
824:
820:
816:
812:
806:
802:
798:
794:
790:
786:
782:
778:
774:
768:
764:
760:
756:
752:
748:
744:
738:
734:
730:
726:
722:
716:
712:
708:
704:
700:
696:
690:
686:
682:
678:
674:
668:
664:
660:
656:
650:
646:
642:
638:
634:
630:
626:
622:
618:
614:
608:
604:
598:
594:
590:
586:
582:
578:
576:
571:
567:
566:is writtten
563:
559:
557:
552:
548:
544:
540:
536:
532:
528:
525:
521:
517:
513:
509:
505:
504:≠ 0 and let
501:
499:
496:Divisibility
490:
480:
476:
472:
468:
464:
462:
407:
405:
359:
355:
351:
347:
343:
339:
337:
290:
286:
280:
276:
269:
252:
250:
240:
236:
232:
228:
224:
218:
214:
210:
202:
198:
194:
192:
188:
183:
179:
175:
171:
167:
163:
157:
153:
149:
145:
141:
137:
135:
132:
127:
123:
119:
115:
109:
105:
101:
97:
93:
90:prime number
85:
83:
80:Formulations
75:
50:
46:
40:
2070:|. That is
1973:) = 1 then
1629:such that
1271:. That is,
1263:imply that
847:> 0 and
524:is said to
2274:, Oxford:
2198:0387962549
2167:References
1791:Dividing
1675:such that
1571:such that
1481:Proof. If
1458:) and let
1364:Lemma. If
1345:such that
1191:such that
1160:) and let
1104:Lemma. If
1085:such that
859:such that
811:Lemma. If
773:Lemma. If
743:Lemma. If
721:Lemma. If
695:Lemma. If
673:Lemma. If
655:Lemma. If
613:Lemma. If
577:Lemma. If
558:Notation.
512:such that
294:such that
2050:|. Since
2022:≠ 0, let
2000:= 0, gcd(
1547:). Since
1493:= gcd(0,
1485:= 0 then
1446:≠ 0. Let
1378:= 0 then
1148:≠ 0. Let
929:≥ 0, and
896:remainder
685:≠ 0 then
603:Lemma. 1|
581:≠ 0 then
2300:Springer
2246:citation
2206:citation
2189:Springer
2090:See also
1996:≠ 0. If
1965:and gcd(
1925:≠ 0 and
1895:≠ 0 and
1757:divides
1531:≠ 0 and
1396:≠ 0 and
1203:, 0 ≤
1144:≠ 0 and
1125:≠ 0 and
1108:≠ 0 and
1065:≠ 0 and
1051:< 0.
949:< 0.
888:quotient
871:and 0 ≤
815:≠ 0 and
562:divides
549:multiple
362:making
182:divides
174:divides
126:divides
118:divides
20: |
2042:) = 1,
1805:gives
1734:, i.e.
1725:gives
1706:gives
1687:gives
1609:i) Let
1470:. Then
1442:≠ 0 or
1423:≠ 0 or
1368:≠ 0 or
1329:≠ 0 a
1325:≠ 0 or
1303:, i.e.
1172:. Then
894:is the
886:is the
645:, and |
607:and −1|
537:divisor
247:History
213:, then
178:, then
152:, then
53:) is a
22:Sandbox
2337:
2306:
2282:
2235:
2195:
2026:= lcm(
1863:. But
1777:= gcd(
1599:; ii)
1539:= lcd(
1501:|, so
1450:= gcd(
1152:= lcm(
1069:≠ 0 a
823:then |
621:then −
526:divide
261:Proofs
201:, and
2102:Notes
1945:) = |
1915:) = |
1907:)gcd(
1785:) ≤ |
1591:then
1497:) = |
1239:that
1207:<
1023:+ 1,
1011:>
959:<
875:<
827:| ≤ |
759:then
711:then
663:then
547:is a
535:is a
114:Then
88:be a
55:lemma
16:<
2335:ISBN
2304:ISBN
2280:ISBN
2258:help
2233:ISBN
2218:help
2193:ISBN
2171:The
2058:and
1859:and
1824:and
1753:and
1717:abte
1713:gmte
1690:eres
1667:and
1625:and
1617:and
1587:and
1555:and
1509:and
1466:and
1412:and
1353:and
1337:and
1279:and
1255:and
1231:and
1187:and
1168:and
1093:and
1077:and
1039:+ 1)
991:− 1)
945:+ 1)
890:and
855:and
793:and
789:and
781:and
751:and
729:and
703:and
681:and
593:and
585:|0,
543:and
358:and
342:and
338:Let
289:and
279:and
170:and
144:and
100:and
84:Let
2078:so
2066:= |
2046:= |
2018:If
1949:|.
1919:|.
1883:|.
1879:= |
1871:≥ |
1851:so
1849:),
1822:),
1801:by
1789:|.
1770:|.
1766:≤ |
1761:,
1749:of
1709:abg
1698:mte
1677:ers
1651:ers
1607:|.
1603:= |
1435:).
1390:|.
1333:of
1137:).
1073:of
1035:− (
999:+ a
987:− (
941:− (
653:|.
649:|||
637:, −
555:.
551:of
539:of
520:,
243:.)
235:or
209:to
205:is
193:If
162:If
136:If
122:or
41:In
34:or
2302:,
2278:,
2250::
2248:}}
2244:{{
2210::
2208:}}
2204:{{
2191:,
2086:.
2076:bc
2072:ab
2068:ab
2052:bc
2048:ab
2038:,
2030:,
2016:.
1992:,
1990:bc
1981:.
1969:,
1963:bc
1947:ab
1941:,
1933:,
1917:ab
1903:,
1837:=
1810:=
1798:gm
1796:=
1794:ab
1781:,
1732:te
1730:=
1723:ab
1719:,
1715:=
1711:=
1700:.
1696:=
1694:ab
1692:=
1681:mt
1679:=
1661:.
1659:br
1657:=
1655:as
1653:=
1646:.
1644:es
1642:=
1638:,
1636:er
1634:=
1577:gm
1575:=
1573:ab
1565:ab
1559:,
1549:ab
1543:,
1525:.
1478:.
1454:,
1431:,
1416:.
1361:.
1311:.
1301:qm
1299:=
1283:.
1223:,
1221:qm
1219:−
1215:=
1199:+
1197:qm
1195:=
1180:.
1156:,
1133:,
1116:ab
1101:.
1047:−
1043:=
1031:≤
1029:ua
1027:−
1019:≥
1001:≥
995:=
983:≥
981:ua
979:−
975::
967:≤
963:,
937:=
933:−
923:qa
921:−
917:=
909::
907:ua
905:−
898:.
879:.
867:+
865:qa
863:=
809:.
807:cy
805:±
803:bx
771:.
767:±
741:.
739:bx
719:.
693:.
691:bc
687:ac
671:.
661:bc
657:ac
641:|−
633:|−
629:,
611:.
601:.
597:|−
574:.
531:;
518:aq
516:=
469:ab
410::
391:1.
352:ab
323:1.
199:ab
186:.
176:ab
158:ab
112:.)
110:ab
73:.
45:,
2260:)
2220:)
2084:c
2082:|
2080:a
2074:|
2064:m
2060:b
2056:a
2044:m
2040:b
2036:a
2032:b
2028:a
2024:m
2020:b
2014:c
2012:|
2010:a
2006:a
2002:a
1998:b
1994:a
1988:|
1986:a
1979:c
1977:|
1975:a
1971:b
1967:a
1961:|
1959:a
1943:b
1939:a
1935:b
1931:a
1927:b
1923:a
1913:b
1911:,
1909:a
1905:b
1901:a
1897:b
1893:a
1881:g
1877:d
1873:g
1869:d
1865:d
1861:b
1857:a
1853:g
1847:a
1845:/
1843:m
1841:(
1839:g
1835:b
1830:b
1828:/
1826:m
1820:b
1818:/
1816:m
1814:(
1812:g
1808:a
1803:b
1787:g
1783:b
1779:a
1775:d
1768:g
1764:e
1759:g
1755:b
1751:a
1747:e
1742:.
1740:g
1738:|
1736:e
1728:g
1704:g
1685:e
1673:t
1669:b
1665:a
1640:b
1632:a
1627:s
1623:r
1619:b
1615:a
1611:e
1605:g
1601:d
1597:g
1595:|
1593:e
1589:b
1585:a
1581:e
1569:g
1563:|
1561:m
1557:b
1553:a
1545:b
1541:a
1537:m
1533:b
1529:a
1523:d
1521:|
1519:k
1515:b
1513:|
1511:k
1507:a
1505:|
1503:k
1499:b
1495:b
1491:d
1487:b
1483:a
1476:d
1474:|
1472:e
1468:b
1464:a
1460:e
1456:b
1452:a
1448:d
1444:b
1440:a
1433:b
1429:a
1425:b
1421:a
1414:b
1410:a
1406:b
1402:a
1398:b
1394:a
1388:b
1384:b
1380:b
1376:a
1370:b
1366:a
1359:b
1357:|
1355:k
1351:a
1349:|
1347:k
1343:k
1339:b
1335:a
1327:b
1323:a
1309:n
1307:|
1305:m
1297:n
1293:r
1289:m
1285:r
1281:b
1277:a
1273:r
1269:r
1267:|
1265:b
1261:n
1259:|
1257:b
1253:m
1251:|
1249:b
1245:r
1243:|
1241:a
1237:n
1235:|
1233:a
1229:m
1227:|
1225:a
1217:n
1213:r
1209:m
1205:r
1201:r
1193:n
1189:r
1185:q
1178:n
1176:|
1174:m
1170:b
1166:a
1162:n
1158:b
1154:a
1150:m
1146:b
1142:a
1135:b
1131:a
1127:b
1123:a
1110:b
1106:a
1099:k
1097:|
1095:b
1091:k
1089:|
1087:a
1083:k
1079:b
1075:a
1067:b
1063:a
1049:a
1045:r
1041:a
1037:q
1033:b
1025:b
1021:q
1017:u
1013:q
1009:u
1005:.
1003:a
997:r
993:a
989:q
985:b
977:b
973:a
969:q
965:u
961:q
957:u
953:q
947:a
943:q
939:b
935:a
931:r
927:r
919:b
915:r
911:u
903:b
892:r
884:q
877:a
873:r
869:r
861:b
857:r
853:q
849:b
845:a
833:b
829:b
825:a
821:b
819:|
817:a
813:b
801:|
799:a
795:y
791:x
787:c
785:|
783:a
779:b
777:|
775:a
769:c
765:b
763:|
761:a
757:c
755:|
753:a
749:b
747:|
745:a
737:|
735:a
731:x
727:b
725:|
723:a
717:c
715:|
713:a
709:c
707:|
705:b
701:b
699:|
697:a
689:|
683:c
679:b
677:|
675:a
669:b
667:|
665:a
659:|
651:b
647:a
643:b
639:a
635:b
631:a
627:b
625:|
623:a
619:b
617:|
615:a
609:a
605:a
599:a
595:a
591:a
589:|
587:a
583:a
579:a
572:b
570:|
568:a
564:b
560:a
553:a
545:b
541:b
533:a
529:b
522:a
514:b
510:q
506:b
502:a
481:n
477:b
473:n
465:n
448:.
445:b
442:=
439:b
436:a
433:s
430:+
427:b
424:n
421:r
408:b
388:=
385:a
382:s
379:+
376:n
373:r
360:s
356:r
350:|
348:n
344:n
340:a
320:=
317:y
314:s
311:+
308:x
305:r
291:s
287:r
281:y
277:x
241:a
237:n
233:a
231:|
229:n
225:n
221:.
219:b
217:|
215:n
211:a
203:n
197:|
195:n
184:b
180:p
172:p
168:a
164:p
160:.
154:p
150:b
146:p
142:a
138:p
128:b
124:p
120:a
116:p
108:|
106:p
102:b
98:a
94:p
86:p
38:.
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