1184:
834:
1179:{\displaystyle {\begin{aligned}\vdots \\12&\times ({\color {blue}{-10}})&+\;\;42&\times \color {blue}{3}&=6\\12&\times ({\color {red}{-3}})&+\;\;42&\times \color {red}{1}&=6\\12&\times \color {red}{4}&+\;\;42&\times ({\color {red}{-1}})&=6\\12&\times \color {blue}{11}&+\;\;42&\times ({\color {blue}{-3}})&=6\\12&\times \color {blue}{18}&+\;\;42&\times ({\color {blue}{-5}})&=6\\\vdots \end{aligned}}}
2508:
On these pages, Bachet proves (without equations) "Proposition XVIII. Deux nombres premiers entre eux estant donnez, treuver le moindre multiple de chascun d’iceux, surpassant de l'unité un multiple de l'autre." (Given two numbers relatively prime, find the lowest multiple of each of them one
505:
1598:
1843:
375:
1454:
2109:
415:
1735:
1475:
839:
1955:
2014:
1470:
2416: – About algebraic curves passing through all intersection points of two other curves, an analogue of Bézout's identity for homogeneous polynomials in three indeterminates
1730:
2501:
617:
591:
565:
539:
654:
306:
829:. Then the following Bézout's identities are had, with the Bézout coefficients written in red for the minimal pairs and in blue for the other ones.
2400:(1730–1783) proved this identity for polynomials. This statement for integers can be found already in the work of an earlier French mathematician,
1399:
2019:
2557:
2205:
exactly in the same ways as for integers. In particular the Bézout's coefficients and the greatest common divisor may be computed with the
2401:
1886:
2138:
2484:
2431:
2617:
2612:
2217:
2274:
2206:
300:
179:
2533:"Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany"
237:
As an example, the greatest common divisor of 15 and 69 is 3, and 3 can be written as a combination of 15 and 69 as
2263:
412:
are both nonzero and none of them divides the other, then exactly two of the pairs of Bézout coefficients satisfy
249:
2520:) is a special case of Bézout's equation and was used by Bachet to solve the problems appearing on pages 199 ff.
1960:
500:{\displaystyle |x|<\left|{\frac {b}{d}}\right|\quad {\text{and}}\quad |y|<\left|{\frac {a}{d}}\right|.}
1340:
84:
2290:
264:
31:
2198:
2577:
30:
This article is about Bézout's theorem in arithmetic. For Bézout's theorem in algebraic geometry, see
2532:
2452:
2419:
2397:
2213:
50:
2358:
2202:
2472:
1385:
664:
267:. Every theorem that results from Bézout's identity is thus true in all principal ideal domains.
1593:{\displaystyle {\begin{aligned}r&=a-qd\\&=a-q(as+bt)\\&=a(1-qs)-bqt.\end{aligned}}}
2585:
2480:
2425:
245:
2547:
2380:
596:
570:
544:
518:
256:
2144:
1838:{\displaystyle {\begin{aligned}d&=as+bt\\&=cus+cvt\\&=c(us+vt).\end{aligned}}}
260:
2216:
of two polynomials are the roots of their greatest common divisor, Bézout's identity and
2143:
Bézout's identity does not always hold for polynomials. For example, when working in the
633:
2413:
2286:
2267:
2139:
Polynomial greatest common divisor § Bézout's identity and extended GCD algorithm
2606:
2588:
2394:
2391:
2273:
The generalization of this result to any number of polynomials and indeterminates is
17:
2456:
803:
The extended
Euclidean algorithm always produces one of these two minimal pairs.
38:
2506:(2nd ed.). Lyons, France: Pierre Rigaud & Associates. pp. 18–33.
2552:
758:
The two pairs of small Bézout's coefficients are obtained from the given one
2593:
263:
in which Bézout's identity holds. In particular, Bézout's identity holds in
178:; they are not unique. A pair of Bézout coefficients can be computed by the
182:, and this pair is, in the case of integers one of the two pairs such that
80:
54:
2285:
As noted in the introduction, Bézout's identity works not only in the
370:{\displaystyle \left(x-k{\frac {b}{d}},\ y+k{\frac {a}{d}}\right),}
2509:
multiple exceeds the other by unity (1).) This problem (namely,
2503:
Problèmes plaisants & délectables qui se font par les nombres
1883:
Bézout's identity can be extended to more than two integers: if
1325:
is a nonempty set of positive integers, it has a minimum element
2379:
An integral domain in which Bézout's identity holds is called a
2428: – A prime divisor of a product divides one of the factors
2164:, but there does not exist any integer-coefficient polynomials
2422: – Polynomial equation whose integer solutions are sought
1449:{\displaystyle a=dq+r\quad {\text{with}}\quad 0\leq r<d.}
2104:{\displaystyle d=a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}}
244:
Many other theorems in elementary number theory, such as
774:
in the above formula either of the two integers next to
2231:
with coefficients in a field, there exist polynomials
287:
are not both zero and one pair of Bézout coefficients
27:
Relating two numbers and their greatest common divisor
2022:
1963:
1889:
1733:
1473:
1402:
837:
636:
599:
573:
547:
521:
418:
309:
2500:Claude Gaspard Bachet (sieur de Méziriac) (1624).
2103:
2008:
1949:
1837:
1592:
1448:
1200:is the original pair of Bézout coefficients, then
1178:
648:
611:
585:
559:
533:
499:
369:
2434: – Integers have unique prime factorizations
1950:{\displaystyle \gcd(a_{1},a_{2},\ldots ,a_{n})=d}
1890:
53:who proved it for polynomials, is the following
2120:is the smallest positive integer of this form
8:
2147:of integers: the greatest common divisor of
303:), all pairs can be represented in the form
2123:every number of this form is a multiple of
656:), then one pair of Bézout coefficients is
61:
2458:Théorie générale des équations algébriques
1128:
1127:
1064:
1063:
1000:
999:
947:
946:
883:
882:
401:, and the fractions simplify to integers.
299:has been computed (for example, using the
2551:
2095:
2085:
2066:
2056:
2043:
2033:
2021:
2009:{\displaystyle x_{1},x_{2},\ldots ,x_{n}}
2000:
1981:
1968:
1962:
1932:
1913:
1900:
1888:
1734:
1732:
1474:
1472:
1422:
1401:
1144:
1142:
1116:
1080:
1078:
1052:
1016:
1014:
988:
959:
928:
926:
895:
864:
862:
838:
836:
635:
598:
572:
546:
520:
480:
468:
460:
454:
439:
427:
419:
417:
349:
324:
308:
1367:, and that for any other common divisor
2444:
2477:Galois' Theory of Algebraic Equations
2197:However, Bézout's identity works for
1296:is nonempty since it contains either
1143:
1115:
1079:
1051:
894:
863:
241:, with Bézout coefficients −9 and 2.
118:. Moreover, the integers of the form
7:
1640:is the smallest positive integer in
138:Here the greatest common divisor of
2289:of integers, but also in any other
2531:Maarten Bullynck (February 2009).
1347:is the greatest common divisor of
1239:(18 − 3 ⋅ 7, −5 + 3 ⋅ 2) = (−3, 1)
1235:(18 − 2 ⋅ 7, −5 + 2 ⋅ 2) = (4, −1)
389:is the greatest common divisor of
25:
2432:Fundamental theorem of arithmetic
2402:Claude Gaspard Bachet de Méziriac
1656:necessarily 0. This implies that
1015:
987:
958:
927:
252:, result from Bézout's identity.
222:; equality occurs only if one of
2563:from the original on 2022-10-09.
2461:. Paris, France: Ph.-D. Pierres.
2317:is a greatest common divisor of
2479:. Singapore: World Scientific.
1427:
1421:
459:
453:
2218:fundamental theorem of algebra
2111:has the following properties:
1938:
1893:
1825:
1807:
1568:
1553:
1537:
1519:
1154:
1139:
1090:
1075:
1026:
1011:
938:
923:
874:
859:
667:: given two non-zero integers
469:
461:
428:
420:
1:
2220:imply the following result:
1219:yields the minimal pairs via
663:This relies on a property of
128:are exactly the multiples of
2207:extended Euclidean algorithm
727:, and another one such that
687:, there is exactly one pair
567:for one of these pairs, and
301:extended Euclidean algorithm
234:is a multiple of the other.
180:extended Euclidean algorithm
92:. Then there exist integers
2281:For principal ideal domains
2262:have no common root in any
2223:For univariate polynomials
1249:Given any nonzero integers
515:are both positive, one has
2634:
2370:is principal and equal to
2325:, then there are elements
2264:algebraically closed field
2136:
1879:For three or more integers
29:
2357:. The reason is that the
2275:Hilbert's Nullstellensatz
1691:be any common divisor of
1355:, it must be proven that
383:is an arbitrary integer,
250:Chinese remainder theorem
2553:10.1016/j.hm.2008.08.009
1957:then there are integers
1648:can therefore not be in
2618:Lemmas in number theory
2266:(commonly the field of
1699:; that is, there exist
1676:is a common divisor of
1359:is a common divisor of
1341:well-ordering principle
265:principal ideal domains
85:greatest common divisor
2580:for Bézout's identity.
2291:principal ideal domain
2199:univariate polynomials
2105:
2010:
1951:
1839:
1594:
1450:
1180:
650:
613:
612:{\displaystyle y>0}
587:
586:{\displaystyle x<0}
561:
560:{\displaystyle y<0}
535:
534:{\displaystyle x>0}
501:
371:
271:Structure of solutions
239:3 = 15 × (−9) + 69 × 2
2613:Diophantine equations
2106:
2011:
1952:
1840:
1668:is also a divisor of
1595:
1451:
1181:
651:
614:
588:
562:
536:
502:
372:
2540:Historia Mathematica
2420:Diophantine equation
2020:
1961:
1887:
1731:
1471:
1400:
835:
634:
630:(including the case
597:
571:
545:
519:
416:
307:
2589:"Bézout's Identity"
2473:Tignol, Jean-Pierre
2293:(PID). That is, if
649:{\displaystyle b=0}
164:Bézout coefficients
65: —
18:Bézout coefficients
2586:Weisstein, Eric W.
2101:
2006:
1947:
1835:
1833:
1590:
1588:
1446:
1396:may be written as
1386:Euclidean division
1176:
1174:
1152:
1121:
1088:
1057:
1024:
993:
964:
936:
900:
872:
665:Euclidean division
646:
619:for the other. If
609:
583:
557:
531:
497:
367:
63:
2578:Online calculator
1425:
488:
457:
447:
357:
339:
332:
62:Bézout's identity
43:Bézout's identity
16:(Redirected from
2625:
2599:
2598:
2565:
2564:
2562:
2555:
2537:
2527:
2521:
2519:
2507:
2497:
2491:
2490:
2469:
2463:
2462:
2449:
2375:
2369:
2356:
2342:
2336:
2330:
2324:
2320:
2316:
2312:
2307:are elements of
2306:
2302:
2298:
2261:
2257:
2253:
2242:
2236:
2230:
2226:
2193:
2175:
2169:
2159:
2153:
2128:
2119:
2110:
2108:
2107:
2102:
2100:
2099:
2090:
2089:
2071:
2070:
2061:
2060:
2048:
2047:
2038:
2037:
2015:
2013:
2012:
2007:
2005:
2004:
1986:
1985:
1973:
1972:
1956:
1954:
1953:
1948:
1937:
1936:
1918:
1917:
1905:
1904:
1869:
1859:
1852:
1849:is a divisor of
1848:
1844:
1842:
1841:
1836:
1834:
1797:
1766:
1726:
1716:
1706:
1702:
1698:
1694:
1690:
1683:
1679:
1675:
1672:, and therefore
1671:
1667:
1663:
1660:is a divisor of
1659:
1655:
1651:
1647:
1644:: the remainder
1643:
1639:
1635:
1624:
1613:
1603:
1599:
1597:
1596:
1591:
1589:
1543:
1503:
1466:
1459:
1455:
1453:
1452:
1447:
1426:
1423:
1395:
1391:
1380:
1370:
1366:
1362:
1358:
1354:
1350:
1346:
1343:. To prove that
1338:
1324:
1320:
1313:
1306:
1299:
1295:
1291:
1256:
1252:
1240:
1236:
1232:
1225:
1218:
1216:
1214:
1213:
1210:
1207:
1199:
1185:
1183:
1182:
1177:
1175:
1153:
1151:
1120:
1089:
1087:
1056:
1025:
1023:
992:
963:
937:
935:
899:
873:
871:
828:
827:gcd (12, 42) = 6
824:
817:
799:
798:
796:
795:
786:
783:
773:
770:by choosing for
769:
754:
748:
740:
726:
724:
712:
698:
686:
683:does not divide
682:
678:
672:
659:
655:
653:
652:
647:
629:
626:is a divisor of
625:
618:
616:
615:
610:
592:
590:
589:
584:
566:
564:
563:
558:
540:
538:
537:
532:
514:
510:
506:
504:
503:
498:
493:
489:
481:
472:
464:
458:
455:
452:
448:
440:
431:
423:
411:
407:
400:
394:
388:
382:
376:
374:
373:
368:
363:
359:
358:
350:
337:
333:
325:
298:
286:
280:
240:
233:
227:
221:
219:
209:
201:
199:
189:
177:
161:
155:
149:
145:
141:
133:
127:
117:
103:
97:
91:
78:
72:
66:
32:Bézout's theorem
21:
2633:
2632:
2628:
2627:
2626:
2624:
2623:
2622:
2603:
2602:
2584:
2583:
2574:
2569:
2568:
2560:
2535:
2530:
2528:
2524:
2510:
2499:
2498:
2494:
2487:
2471:
2470:
2466:
2451:
2450:
2446:
2441:
2410:
2389:
2371:
2361:
2344:
2338:
2332:
2326:
2322:
2318:
2314:
2308:
2304:
2300:
2294:
2283:
2271:
2268:complex numbers
2259:
2255:
2254:if and only if
2244:
2238:
2232:
2228:
2224:
2177:
2171:
2165:
2155:
2148:
2145:polynomial ring
2141:
2135:
2133:For polynomials
2124:
2115:
2091:
2081:
2062:
2052:
2039:
2029:
2018:
2017:
1996:
1977:
1964:
1959:
1958:
1928:
1909:
1896:
1885:
1884:
1881:
1876:
1874:Generalizations
1861:
1860:, this implies
1854:
1850:
1846:
1832:
1831:
1795:
1794:
1764:
1763:
1741:
1729:
1728:
1727:. One has thus
1718:
1708:
1704:
1700:
1696:
1692:
1688:
1681:
1677:
1673:
1669:
1665:
1661:
1657:
1653:
1649:
1645:
1641:
1637:
1626:
1615:
1605:
1604:is of the form
1601:
1587:
1586:
1541:
1540:
1501:
1500:
1481:
1469:
1468:
1461:
1457:
1398:
1397:
1393:
1389:
1372:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1326:
1322:
1315:
1308:
1301:
1297:
1293:
1258:
1254:
1250:
1247:
1245:Existence proof
1238:
1234:
1227:
1226:, respectively
1220:
1211:
1208:
1205:
1204:
1202:
1201:
1189:
1173:
1172:
1166:
1165:
1157:
1132:
1122:
1108:
1102:
1101:
1093:
1068:
1058:
1044:
1038:
1037:
1029:
1004:
994:
980:
974:
973:
965:
951:
941:
916:
910:
909:
901:
887:
877:
852:
846:
845:
833:
832:
826:
819:
812:
809:
787:
784:
779:
778:
776:
775:
771:
759:
744:
742:
728:
720:
714:
700:
688:
684:
680:
674:
668:
657:
632:
631:
627:
620:
595:
594:
569:
568:
543:
542:
517:
516:
512:
508:
476:
435:
414:
413:
409:
405:
396:
390:
384:
378:
314:
310:
305:
304:
288:
282:
276:
273:
261:integral domain
238:
229:
223:
211:
210:| ≤ |
205:
203:
191:
190:| ≤ |
185:
183:
167:
157:
151:
150:. The integers
147:
146:is taken to be
143:
139:
136:
129:
119:
105:
99:
93:
87:
74:
68:
64:
49:), named after
35:
28:
23:
22:
15:
12:
11:
5:
2631:
2629:
2621:
2620:
2615:
2605:
2604:
2601:
2600:
2581:
2573:
2572:External links
2570:
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2566:
2522:
2492:
2485:
2464:
2443:
2442:
2440:
2437:
2436:
2435:
2429:
2426:Euclid's lemma
2423:
2417:
2409:
2406:
2398:Étienne Bézout
2388:
2385:
2299:is a PID, and
2282:
2279:
2222:
2212:As the common
2137:Main article:
2134:
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1477:
1476:
1456:The remainder
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840:
808:
805:
645:
642:
639:
608:
605:
602:
582:
579:
576:
556:
553:
550:
530:
527:
524:
496:
492:
487:
484:
479:
475:
471:
467:
463:
451:
446:
443:
438:
434:
430:
426:
422:
366:
362:
356:
353:
348:
345:
342:
336:
331:
328:
323:
320:
317:
313:
272:
269:
246:Euclid's lemma
59:
51:Étienne Bézout
47:Bézout's lemma
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2630:
2619:
2616:
2614:
2611:
2610:
2608:
2596:
2595:
2590:
2587:
2582:
2579:
2576:
2575:
2571:
2559:
2554:
2549:
2545:
2541:
2534:
2526:
2523:
2517:
2513:
2505:
2504:
2496:
2493:
2488:
2486:981-02-4541-6
2482:
2478:
2474:
2468:
2465:
2460:
2459:
2454:
2448:
2445:
2438:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2414:AF+BG theorem
2412:
2411:
2407:
2405:
2404:(1581–1638).
2403:
2399:
2396:
2395:mathematician
2393:
2386:
2384:
2382:
2381:Bézout domain
2377:
2374:
2368:
2364:
2360:
2355:
2351:
2347:
2341:
2335:
2329:
2311:
2297:
2292:
2288:
2280:
2278:
2276:
2269:
2265:
2251:
2247:
2241:
2235:
2221:
2219:
2215:
2210:
2208:
2204:
2200:
2195:
2192:
2188:
2185:
2181:
2174:
2168:
2163:
2158:
2152:
2146:
2140:
2132:
2127:
2122:
2118:
2114:
2113:
2112:
2096:
2092:
2086:
2082:
2078:
2075:
2072:
2067:
2063:
2057:
2053:
2049:
2044:
2040:
2034:
2030:
2026:
2023:
2001:
1997:
1993:
1990:
1987:
1982:
1978:
1974:
1969:
1965:
1944:
1941:
1933:
1929:
1925:
1922:
1919:
1914:
1910:
1906:
1901:
1897:
1878:
1873:
1871:
1868:
1864:
1857:
1828:
1822:
1819:
1816:
1813:
1810:
1804:
1801:
1799:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1768:
1760:
1757:
1754:
1751:
1748:
1745:
1743:
1738:
1725:
1721:
1715:
1711:
1685:
1634:
1630:
1622:
1618:
1612:
1608:
1583:
1580:
1577:
1574:
1571:
1565:
1562:
1559:
1556:
1550:
1547:
1545:
1534:
1531:
1528:
1525:
1522:
1516:
1513:
1510:
1507:
1505:
1497:
1494:
1491:
1488:
1485:
1483:
1478:
1464:
1443:
1440:
1437:
1434:
1431:
1428:
1418:
1415:
1412:
1409:
1406:
1403:
1387:
1382:
1379:
1375:
1342:
1337:
1333:
1329:
1318:
1311:
1305:
1289:
1285:
1281:
1277:
1273:
1269:
1265:
1261:
1244:
1242:
1230:
1223:
1197:
1193:
1186:
1169:
1162:
1159:
1148:
1145:
1136:
1134:
1129:
1124:
1117:
1112:
1110:
1105:
1098:
1095:
1084:
1081:
1072:
1070:
1065:
1060:
1053:
1048:
1046:
1041:
1034:
1031:
1020:
1017:
1008:
1006:
1001:
996:
989:
984:
982:
977:
970:
967:
960:
955:
953:
948:
943:
932:
929:
920:
918:
913:
906:
903:
896:
891:
889:
884:
879:
868:
865:
856:
854:
849:
842:
830:
822:
815:
806:
804:
801:
794:
790:
782:
767:
763:
756:
752:
747:
739:
735:
731:
723:
718:
711:
707:
703:
696:
692:
677:
671:
666:
661:
643:
640:
637:
623:
606:
603:
600:
580:
577:
574:
554:
551:
548:
528:
525:
522:
494:
490:
485:
482:
477:
473:
465:
449:
444:
441:
436:
432:
424:
402:
399:
393:
387:
381:
364:
360:
354:
351:
346:
343:
340:
334:
329:
326:
321:
318:
315:
311:
302:
296:
292:
285:
279:
270:
268:
266:
262:
258:
257:Bézout domain
253:
251:
247:
242:
235:
232:
226:
218:
214:
208:
198:
194:
188:
181:
175:
171:
165:
160:
154:
135:
132:
126:
122:
116:
112:
108:
102:
96:
90:
86:
82:
77:
71:
58:
56:
52:
48:
45:(also called
44:
40:
33:
19:
2592:
2546:(1): 48–72.
2543:
2539:
2525:
2515:
2511:
2502:
2495:
2476:
2467:
2457:
2447:
2390:
2378:
2372:
2366:
2362:
2353:
2349:
2345:
2339:
2333:
2327:
2309:
2295:
2284:
2272:
2249:
2245:
2239:
2233:
2211:
2196:
2190:
2186:
2183:
2179:
2172:
2166:
2161:
2156:
2150:
2142:
2125:
2116:
1882:
1866:
1862:
1855:
1723:
1719:
1713:
1709:
1686:
1664:. Similarly
1632:
1628:
1620:
1616:
1614:, and hence
1610:
1606:
1462:
1383:
1377:
1373:
1335:
1331:
1327:
1316:
1309:
1303:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1259:
1248:
1228:
1221:
1198:) = (18, −5)
1195:
1191:
1187:
831:
820:
813:
810:
802:
792:
788:
780:
765:
761:
757:
750:
749:| <
745:
737:
733:
729:
721:
716:
709:
705:
701:
694:
690:
675:
669:
662:
621:
403:
397:
391:
385:
379:
294:
290:
283:
277:
274:
254:
243:
236:
230:
224:
216:
212:
206:
196:
192:
186:
173:
169:
163:
158:
152:
137:
130:
124:
120:
114:
110:
106:
100:
94:
88:
75:
69:
60:
46:
42:
36:
2176:satisfying
1625:. However,
1233:; that is,
719:< |
162:are called
39:mathematics
2607:Categories
2529:See also:
2453:Bézout, É.
2343:such that
2243:such that
2016:such that
1707:such that
1467:, because
1371:, one has
1292:. The set
699:such that
104:such that
2594:MathWorld
2076:⋯
1991:…
1923:…
1845:That is,
1687:Now, let
1652:, making
1572:−
1560:−
1514:−
1492:−
1432:≤
1339:, by the
1321:). Since
1170:⋮
1146:−
1137:×
1113:×
1082:−
1073:×
1049:×
1018:−
1009:×
985:×
956:×
930:−
921:×
892:×
866:−
857:×
843:⋮
319:−
2558:Archived
2475:(2001).
2455:(1779).
2408:See also
1853:. Since
81:integers
2387:History
2201:over a
1290:> 0}
1215:
1203:
825:, then
807:Example
797:
777:
743:−|
715:0 <
248:or the
55:theorem
2483:
2392:French
2313:, and
1858:> 0
1636:, and
1460:is in
1307:(with
1257:, let
1237:, and
753:< 0
725:|
658:(1, 0)
624:> 0
377:where
338:
259:is an
220:|
204:|
200:|
184:|
2561:(PDF)
2536:(PDF)
2439:Notes
2359:ideal
2214:roots
2203:field
1631:<
1623:∪ {0}
1600:Thus
1465:∪ {0}
679:, if
83:with
2481:ISBN
2331:and
2321:and
2303:and
2287:ring
2258:and
2237:and
2227:and
2170:and
2154:and
1717:and
1703:and
1695:and
1680:and
1627:0 ≤
1438:<
1424:with
1384:The
1363:and
1351:and
1314:and
1312:= ±1
1282:and
1253:and
1212:42/6
823:= 42
818:and
816:= 12
811:Let
741:and
713:and
673:and
604:>
593:and
578:<
552:<
541:and
526:>
511:and
474:<
433:<
408:and
395:and
281:and
228:and
202:and
166:for
156:and
142:and
98:and
73:and
67:Let
2548:doi
2518:= 1
2337:in
2252:= 1
2160:is
1891:gcd
1392:by
1388:of
1319:= 0
1300:or
1262:= {
1231:= 3
1224:= 2
1188:If
507:If
456:and
404:If
275:If
79:be
37:In
2609::
2591:.
2556:.
2544:36
2542:.
2538:.
2516:by
2514:−
2512:ax
2383:.
2376:.
2373:Rd
2367:Rb
2365:+
2363:Ra
2352:=
2350:by
2348:+
2346:ax
2277:.
2270:).
2250:bg
2248:+
2246:af
2209:.
2194:.
2189:=
2182:+
2180:xp
1870:.
1865:≤
1724:cv
1722:=
1714:cu
1712:=
1684:.
1619:∈
1611:by
1609:+
1607:ax
1381:.
1376:≤
1336:bt
1334:+
1332:as
1330:=
1288:by
1286:+
1284:ax
1278:∈
1274:,
1270:|
1268:by
1266:+
1264:ax
1241:.
1217:∈
1206:18
1194:,
1130:42
1118:18
1106:12
1066:42
1054:11
1042:12
1002:42
978:12
949:42
914:12
885:42
869:10
850:12
800:.
764:,
755:.
736:+
734:dq
732:=
708:+
706:dq
704:=
693:,
660:.
293:,
255:A
172:,
134:.
125:bt
123:+
121:az
113:=
111:by
109:+
107:ax
57::
41:,
2597:.
2550::
2489:.
2354:d
2340:R
2334:y
2328:x
2323:b
2319:a
2315:d
2310:R
2305:b
2301:a
2296:R
2260:g
2256:f
2240:b
2234:a
2229:g
2225:f
2191:x
2187:q
2184:x
2178:2
2173:q
2167:p
2162:x
2157:x
2151:x
2149:2
2126:d
2117:d
2097:n
2093:x
2087:n
2083:a
2079:+
2073:+
2068:2
2064:x
2058:2
2054:a
2050:+
2045:1
2041:x
2035:1
2031:a
2027:=
2024:d
2002:n
1998:x
1994:,
1988:,
1983:2
1979:x
1975:,
1970:1
1966:x
1945:d
1942:=
1939:)
1934:n
1930:a
1926:,
1920:,
1915:2
1911:a
1907:,
1902:1
1898:a
1894:(
1867:d
1863:c
1856:d
1851:d
1847:c
1829:.
1826:)
1823:t
1820:v
1817:+
1814:s
1811:u
1808:(
1805:c
1802:=
1792:t
1789:v
1786:c
1783:+
1780:s
1777:u
1774:c
1771:=
1761:t
1758:b
1755:+
1752:s
1749:a
1746:=
1739:d
1720:b
1710:a
1705:v
1701:u
1697:b
1693:a
1689:c
1682:b
1678:a
1674:d
1670:b
1666:d
1662:a
1658:d
1654:r
1650:S
1646:r
1642:S
1638:d
1633:d
1629:r
1621:S
1617:r
1602:r
1584:.
1581:t
1578:q
1575:b
1569:)
1566:s
1563:q
1557:1
1554:(
1551:a
1548:=
1538:)
1535:t
1532:b
1529:+
1526:s
1523:a
1520:(
1517:q
1511:a
1508:=
1498:d
1495:q
1489:a
1486:=
1479:r
1463:S
1458:r
1444:.
1441:d
1435:r
1429:0
1419:r
1416:+
1413:q
1410:d
1407:=
1404:a
1394:d
1390:a
1378:d
1374:c
1369:c
1365:b
1361:a
1357:d
1353:b
1349:a
1345:d
1328:d
1323:S
1317:y
1310:x
1304:a
1302:–
1298:a
1294:S
1280:Z
1276:y
1272:x
1260:S
1255:b
1251:a
1229:k
1222:k
1209:/
1196:y
1192:x
1190:(
1163:6
1160:=
1155:)
1149:5
1140:(
1125:+
1099:6
1096:=
1091:)
1085:3
1076:(
1061:+
1035:6
1032:=
1027:)
1021:1
1012:(
997:+
990:4
971:6
968:=
961:1
944:+
939:)
933:3
924:(
907:6
904:=
897:3
880:+
875:)
860:(
821:b
814:a
793:d
791:/
789:b
785:/
781:x
772:k
768:)
766:y
762:x
760:(
751:r
746:d
738:r
730:c
722:d
717:r
710:r
702:c
697:)
695:r
691:q
689:(
685:c
681:d
676:d
670:c
644:0
641:=
638:b
628:b
622:a
607:0
601:y
581:0
575:x
555:0
549:y
529:0
523:x
513:b
509:a
495:.
491:|
486:d
483:a
478:|
470:|
466:y
462:|
450:|
445:d
442:b
437:|
429:|
425:x
421:|
410:b
406:a
398:b
392:a
386:d
380:k
365:,
361:)
355:d
352:a
347:k
344:+
341:y
335:,
330:d
327:b
322:k
316:x
312:(
297:)
295:y
291:x
289:(
284:b
278:a
231:b
225:a
217:d
215:/
213:a
207:y
197:d
195:/
193:b
187:x
176:)
174:b
170:a
168:(
159:y
153:x
148:0
144:0
140:0
131:d
115:d
101:y
95:x
89:d
76:b
70:a
34:.
20:)
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