1286:. Any musical tone can be broken into its fundamental frequency and harmonic frequencies, which are integer multiples of the fundamental. This series is conjectured to be the basis of natural harmony and melody. The tonal complexity of ratios between these harmonics is said to get more complex with higher prime factors. To limit this tonal complexity, an interval is said to be
635:-smooth squarefree number, 2): 1, 3, 5, 6, 10, 15, and 30, each of which leads to a Pell equation. The number of solutions per Pell equation required by Lehmer's method is max(3, (5 + 1)/2) = 3, so this method generates three solutions to each Pell equation, as follows.
1327:). That is, the only pairs of consecutive integers that have only powers of two and three in their prime factorizations are (1,2), (2,3), (3,4), and (8,9). If this is extended to the 5-limit, six additional superparticular ratios are available: 5/4 (the
270:
1269:
describe a computational procedure that, empirically, finds many but not all of the consecutive pairs of smooth numbers described by Størmer's theorem, and is much faster than using Pell's equation to find all solutions.
1202:
1068:
120:
2060:
1982:
1252:> 3, has a prime factor greater than or equal to 137. Størmer's theorem is an important part of his proof, in which he reduces the problem to the solution of 128 Pell equations.
136:
1891:
554:
396:
872:
primes is at most 3 − 2. Lehmer's result produces a tighter bound for sets of small primes: (2 − 1) × max(3,(
1282:, musical intervals can be described as ratios between positive integers. More specifically, they can be described as ratios between members of the
1584:
868:
Størmer's original result can be used to show that the number of consecutive pairs of integers that are smooth with respect to a set of
947:
933:
919:
897:
1105:
974:
1307:
Størmer's theorem allows all possible superparticular ratios in a given limit to be found. For example, in the 3-limit (
2145:
926:
OEIS also lists the number of pairs of this type where the larger of the two integers in the pair is square (sequence
1828:
1304:
are very important in just tuning theory as they represent ratios between adjacent members of the harmonic series.
57:
40:
43:
that there are only a finite number of pairs of this type, but Størmer gave a procedure for finding them all.
2140:
2093:
1987:
1909:
1662:
1255:
Several authors have extended Størmer's work by providing methods for listing the solutions to more general
265:{\displaystyle \left\{p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{k}^{e_{k}}\mid e_{i}\in \{0,1,2,\ldots \}\right\}}
851:= 1 are (31,4), (1921,248), and (119071,15372). The Pell solution (31,4) leads to the pair of consecutive
1542:
1844:
747:= 1 are (19,6), (721,228), and (27379,8658). The Pell solution (19,6) leads to the pair of consecutive
507:
349:
35:
that exist, for a given degree of smoothness, and provides a method for finding all such pairs using
828:= 1 are (11,2), (241,44), and (5291,966). The Pell solution (11,2) leads to the pair of consecutive
672: + 1)/2 for smoothness; the three pairs to be tested are (1,2), (8,9), and (49,50). Both
797:= 1 are (9,2), (161,36), and (2889,646). The Pell solution (9,2) leads to the pair of consecutive
1336:
1283:
1227:
1215:
wrote about this result, saying that it "is very pretty, and there are many applications of it."
774:= 1 are (7,2), (97,28), and (1351,390). The Pell solution (7,2) leads to the pair of consecutive
302:, in each one finding only the smallest solution. A simplified version of the procedure, due to
1820:
1301:
612:
2122:
2075:
1800:
1779:
1750:
1729:
1691:
1652:
1613:
1534:
1256:
306:, is described below; it solves fewer equations but finds more solutions in each equation.
8:
706:= 5, 49, and 485 Lehmer's method forms the three candidate pairs of consecutive numbers (
2110:
1679:
1630:
1601:
1562:
1308:
406:
299:
1813:
1737:
Luo, Jia Gui (1991). "A generalization of the Störmer theorem and some applications".
882:
The number of consecutive pairs of integers that are smooth with respect to the first
287:-smooth numbers. Further, it gives a method of finding them all using Pell equations.
1824:
937:
1660:
Halsey, G. D.; Hewitt, Edwin (1972). "More on the superparticular ratios in music".
2102:
1715:
1671:
1640:
1593:
1554:
1344:
725:
501:
Lehmer's paper furthermore shows that applying a similar procedure to the equation
1815:
Genesis of a Music: An
Account of a Creative Work, Its Roots, and Its Fulfillments
1787:
Mei, Han Fei; Sun, Sheng Fang (1997). "A further extension of Störmer's theorem".
2118:
2071:
1796:
1775:
1746:
1725:
1687:
1648:
1644:
1609:
1530:
1279:
343:-smooth numbers would be odd. Lehmer's method involves solving the Pell equation
1838:
28:
1621:
Conrey, J. B.; Holmstrom, M. A.; McLaughlin, T. L. (2013). "Smooth neighbors".
1348:
1320:
810:
779:
604:
32:
2134:
1720:
1703:
1316:
1294:
1235:
1231:
1212:
721:
318:
36:
20:
1808:
1699:
1558:
1340:
1260:
608:
494:. Thus one can find all such pairs by testing the numbers of this form for
303:
123:
1332:
1328:
1287:
833:
802:
616:
651:= 1 are (3,2), (17,12), and (99,70). Thus, for each of the three values
2114:
1683:
1605:
1324:
752:
677:
52:
1566:
2106:
1675:
1597:
856:
684:-smooth numbers, but (49,50) is not, as 49 has 7 as a prime factor.
1635:
953:
The size of the solutions can also be bounded: in the case where
805:
and the Pell solution (161,36) leads to the pair of consecutive
567:-smooth square-free numbers other than 1, yields those pairs of
1312:
673:
295:
Størmer's original procedure involves solving a set of roughly
1906:
Sun, Qi; Yuan, Ping Zhi (1989). "On the
Diophantine equations
1234:(other than 8,9) in the case where one of the two powers is a
1197:{\displaystyle \log(x-1)<M(2+\log(4S)){\sqrt {S}}-\log(2).}
912:
2, 9, 81, 4375, 9801, 123201, 336141, 11859211, ... (sequence
755:; the other two solutions to the Pell equation do not lead to
1063:{\displaystyle \log(x)<M(2+\log(8S)){\sqrt {2S}}-\log(4),}
699:= 1 are (5,2), (49,20), and (485,198). From the three values
31:, gives a finite bound on the number of consecutive pairs of
1895:
Skrifter
Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl
720: + 1)/2: (2,3), (24,25), and (242,243). Of these,
942:
928:
914:
892:
16:
Gives a finite bound on pairs of consecutive smooth numbers
1420:
1408:
1384:
1372:
571:-smooth numbers separated by 2: the smooth pairs are then
279:. Then Størmer's theorem states that, for every choice of
1620:
1311:), the only possible superparticular ratios are 2/1 (the
1266:
1486:
890:
1, 4, 10, 23, 40, 68, 108, 167, 241, 345, ... (sequence
1789:
Journal of Jishou
University (Natural Science Edition)
313:
be the given set of primes, and define a number to be
1990:
1912:
1847:
1108:
977:
843:= 30, the first three solutions to the Pell equation
820:= 15, the first three solutions to the Pell equation
789:= 10, the first three solutions to the Pell equation
510:
428:
Pell equations to solve. For each such equation, let
352:
139:
60:
1547:
Biographical
Memoirs of Fellows of the Royal Society
766:= 6, the first three solutions to the Pell equation
739:= 5, the first three solutions to the Pell equation
691:= 3, the first three solutions to the Pell equation
643:= 1, the first three solutions to the Pell equation
283:, there are only finitely many pairs of consecutive
2083:Walker, D. T. (1967). "On the diophantine equation
1841:(1897). "Quelques théorèmes sur l'équation de Pell
1545:(1958). "Fredrik Carl Mulertz Stormer, 1874-1957".
1509:
1207:
904:The largest integer from all these pairs, for each
130:-smooth numbers are defined as the set of integers
2054:
1976:
1885:
1812:
1360:
1196:
1062:
548:
390:
264:
114:
1819:(2nd ed.). New York: Da Capo Press. p.
1474:
1396:
658:= 3, 17, and 99, Lehmer's method tests the pair (
2132:
1585:Proceedings of the American Mathematical Society
472:Then, as Lehmer shows, all consecutive pairs of
275:that can be generated by products of numbers in
863:
1263:criteria for the solutions to Pell equations.
1574:Chein, E. Z. (1976). "A note on the equation
115:{\displaystyle P=\{p_{1},p_{2},\dots p_{k}\}}
1659:
1492:
1293:when both its numerator and denominator are
254:
230:
109:
67:
1092:, and in the case where the smooth pair is
950:), as both types of pair arise frequently.
335:; otherwise there could be no consecutive
1719:
1634:
1267:Conrey, Holmstrom & McLaughlin (2013)
420:is generated as a product of a subset of
1757:
1241:
2055:{\displaystyle (ax^{n}+1)/(ax+1)=y^{2}}
1977:{\displaystyle (ax^{n}-1)/(ax-1)=y^{2}}
1905:
1837:
1541:
1464:
1439:
1366:
2133:
2082:
1807:
1786:
1698:
1480:
1468:
1460:
1426:
1414:
1402:
1390:
1378:
1573:
1223:
732:-smooth numbers but (242,243) is not.
603:To find the ten consecutive pairs of
1758:Mabkhout, M. (1993). "Minoration de
1768:Rend. Sem. Fac. Sci. Univ. Cagliari
1736:
1508:
1456:
1452:
1230:on the nonexistence of consecutive
321:if all its prime factors belong to
13:
1886:{\displaystyle x^{2}-Dy^{2}=\pm 1}
1273:
1088:is the product of all elements of
14:
2157:
1351:). All are musically meaningful.
1218:
1208:Generalizations and applications
476:-smooth numbers are of the form
465:is the largest of the primes in
441:be the generated solutions, for
290:
1708:Illinois Journal of Mathematics
1259:, or by providing more general
1226:used Størmer's method to prove
549:{\displaystyle x^{2}-Dy^{2}=1,}
391:{\displaystyle x^{2}-2qy^{2}=1}
2036:
2021:
2013:
1991:
1958:
1943:
1935:
1913:
1445:
1432:
1188:
1182:
1163:
1160:
1151:
1136:
1127:
1115:
1054:
1048:
1026:
1023:
1014:
999:
990:
984:
1:
2094:American Mathematical Monthly
1663:American Mathematical Monthly
1501:
339:-smooth numbers, because all
1645:10.1080/10586458.2013.768483
864:Number and size of solutions
595:solutions of that equation.
46:
7:
1278:In the musical practice of
627:-smooth squarefree numbers
623:= {2,3,5}. There are seven
10:
2162:
1493:Halsey & Hewitt (1972)
713: − 1)/2, (
665: − 1)/2, (
598:
2146:Theorems in number theory
1704:"On a Problem of Størmer"
1244:proved that every number
728:are pairs of consecutive
680:are pairs of consecutive
1893:et leurs applications".
1623:Experimental Mathematics
1354:
41:Thue–Siegel–Roth theorem
445:in the range from 1 to
2056:
1978:
1887:
1721:10.1215/ijm/1256067456
1559:10.1098/rsbm.1958.0021
1302:superparticular ratios
1198:
1064:
613:superparticular ratios
605:{2,3,5}-smooth numbers
550:
392:
266:
116:
39:. It follows from the
2057:
1979:
1888:
1465:Sun & Yuan (1989)
1257:diophantine equations
1199:
1065:
631:(omitting the eighth
551:
393:
267:
117:
2141:Mathematics of music
2064:Sichuan Daxue Xuebao
1988:
1910:
1845:
1739:Sichuan Daxue Xuebao
1461:Mei & Sun (1997)
1228:Catalan's conjecture
1106:
975:
587:is one of the first
573:(x − 1, x + 1)
508:
350:
137:
58:
964:are required to be
456:(inclusive), where
416:. Each such number
213:
188:
166:
2052:
1974:
1883:
1451:In particular see
1347:), and 81/80 (the
1309:Pythagorean tuning
1194:
1060:
546:
407:square-free number
388:
262:
192:
167:
145:
112:
1670:(10): 1096–1100.
1523:Chinese Sci. Bull
1171:
1037:
51:If one chooses a
25:Størmer's theorem
2153:
2126:
2079:
2061:
2059:
2058:
2053:
2051:
2050:
2020:
2006:
2005:
1983:
1981:
1980:
1975:
1973:
1972:
1942:
1928:
1927:
1902:
1892:
1890:
1889:
1884:
1873:
1872:
1857:
1856:
1834:
1818:
1804:
1783:
1754:
1733:
1723:
1695:
1656:
1638:
1617:
1570:
1538:
1496:
1490:
1484:
1478:
1472:
1449:
1443:
1436:
1430:
1424:
1418:
1412:
1406:
1400:
1394:
1388:
1382:
1376:
1370:
1364:
1323:), and 9/8 (the
1203:
1201:
1200:
1195:
1172:
1167:
1098:
1091:
1087:
1083:
1069:
1067:
1066:
1061:
1038:
1030:
967:
963:
956:
945:
931:
917:
895:
855:-smooth numbers
832:-smooth numbers
809:-smooth numbers
801:-smooth numbers
778:-smooth numbers
751:-smooth numbers
594:
586:
574:
570:
566:
563:ranges over all
562:
555:
553:
552:
547:
536:
535:
520:
519:
497:
493:
475:
468:
464:
455:
444:
440:
427:
423:
419:
415:
411:
404:
397:
395:
394:
389:
381:
380:
362:
361:
342:
338:
334:
324:
316:
312:
298:
286:
282:
278:
271:
269:
268:
263:
261:
257:
226:
225:
212:
211:
210:
200:
187:
186:
185:
175:
165:
164:
163:
153:
129:
121:
119:
118:
113:
108:
107:
92:
91:
79:
78:
2161:
2160:
2156:
2155:
2154:
2152:
2151:
2150:
2131:
2130:
2129:
2107:10.2307/2314877
2046:
2042:
2016:
2001:
1997:
1989:
1986:
1985:
1968:
1964:
1938:
1923:
1919:
1911:
1908:
1907:
1868:
1864:
1852:
1848:
1846:
1843:
1842:
1831:
1676:10.2307/2317424
1598:10.2307/2041579
1543:Chapman, Sydney
1504:
1499:
1491:
1487:
1479:
1475:
1450:
1446:
1437:
1433:
1425:
1421:
1413:
1409:
1401:
1397:
1389:
1385:
1377:
1373:
1365:
1361:
1357:
1300:. Furthermore,
1284:harmonic series
1280:just intonation
1276:
1274:In music theory
1242:Mabkhout (1993)
1221:
1210:
1166:
1107:
1104:
1103:
1093:
1089:
1085:
1074:
1029:
976:
973:
972:
965:
958:
954:
941:
927:
913:
891:
877:
866:
718:
711:
704:
670:
663:
656:
601:
588:
576:
572:
568:
564:
560:
531:
527:
515:
511:
509:
506:
505:
495:
490:
485:− 1)/2, (
483:
477:
473:
466:
462:
457:
452:
446:
442:
438:
434:
429:
425:
424:, so there are
421:
417:
413:
409:
402:
376:
372:
357:
353:
351:
348:
347:
340:
336:
332:
326:
322:
314:
310:
296:
293:
284:
280:
276:
221:
217:
206:
202:
201:
196:
181:
177:
176:
171:
159:
155:
154:
149:
144:
140:
138:
135:
134:
127:
103:
99:
87:
83:
74:
70:
59:
56:
55:
49:
17:
12:
11:
5:
2159:
2149:
2148:
2143:
2128:
2127:
2101:(5): 504–513.
2080:
2049:
2045:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2019:
2015:
2012:
2009:
2004:
2000:
1996:
1993:
1971:
1967:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1941:
1937:
1934:
1931:
1926:
1922:
1918:
1915:
1903:
1882:
1879:
1876:
1871:
1867:
1863:
1860:
1855:
1851:
1835:
1829:
1805:
1791:(in Chinese).
1784:
1774:(2): 135–148.
1755:
1745:(4): 469–474.
1734:
1696:
1657:
1629:(2): 195–202.
1618:
1571:
1539:
1529:(4): 275–278.
1505:
1503:
1500:
1498:
1497:
1485:
1473:
1444:
1440:Chapman (1958)
1431:
1419:
1407:
1395:
1383:
1371:
1367:Størmer (1897)
1358:
1356:
1353:
1349:syntonic comma
1345:minor semitone
1343:), 25/24 (the
1339:), 16/15 (the
1321:perfect fourth
1275:
1272:
1232:perfect powers
1220:
1219:In mathematics
1217:
1209:
1206:
1205:
1204:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1170:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1078:= max(3, (max(
1071:
1070:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1036:
1033:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
968:-smooth, then
924:
923:
902:
901:
875:
865:
862:
861:
860:
837:
814:
783:
760:
759:-smooth pairs.
733:
716:
709:
702:
685:
668:
661:
654:
600:
597:
557:
556:
545:
542:
539:
534:
530:
526:
523:
518:
514:
488:
481:
460:
450:
436:
432:
399:
398:
387:
384:
379:
375:
371:
368:
365:
360:
356:
330:
300:Pell equations
292:
289:
273:
272:
260:
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
224:
220:
216:
209:
205:
199:
195:
191:
184:
180:
174:
170:
162:
158:
152:
148:
143:
111:
106:
102:
98:
95:
90:
86:
82:
77:
73:
69:
66:
63:
48:
45:
37:Pell equations
33:smooth numbers
27:, named after
15:
9:
6:
4:
3:
2:
2158:
2147:
2144:
2142:
2139:
2138:
2136:
2124:
2120:
2116:
2112:
2108:
2104:
2100:
2096:
2095:
2090:
2086:
2081:
2077:
2073:
2069:
2065:
2047:
2043:
2039:
2033:
2030:
2027:
2024:
2017:
2010:
2007:
2002:
1998:
1994:
1969:
1965:
1961:
1955:
1952:
1949:
1946:
1939:
1932:
1929:
1924:
1920:
1916:
1904:
1900:
1896:
1880:
1877:
1874:
1869:
1865:
1861:
1858:
1853:
1849:
1840:
1839:Størmer, Carl
1836:
1832:
1830:0-306-71597-X
1826:
1822:
1817:
1816:
1810:
1809:Partch, Harry
1806:
1802:
1798:
1794:
1790:
1785:
1781:
1777:
1773:
1769:
1765:
1761:
1756:
1752:
1748:
1744:
1740:
1735:
1731:
1727:
1722:
1717:
1713:
1709:
1705:
1701:
1700:Lehmer, D. H.
1697:
1693:
1689:
1685:
1681:
1677:
1673:
1669:
1665:
1664:
1658:
1654:
1650:
1646:
1642:
1637:
1632:
1628:
1624:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1586:
1581:
1577:
1572:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1528:
1524:
1520:
1516:
1512:
1507:
1506:
1494:
1489:
1482:
1481:Partch (1974)
1477:
1470:
1469:Walker (1967)
1466:
1462:
1458:
1454:
1448:
1441:
1438:As quoted by
1435:
1428:
1427:Lehmer (1964)
1423:
1416:
1415:Lehmer (1964)
1411:
1404:
1403:Lehmer (1964)
1399:
1392:
1391:Lehmer (1964)
1387:
1380:
1379:Lehmer (1964)
1375:
1368:
1363:
1359:
1352:
1350:
1346:
1342:
1338:
1335:), 10/9 (the
1334:
1330:
1326:
1322:
1318:
1317:perfect fifth
1314:
1310:
1305:
1303:
1299:
1297:
1292:
1290:
1285:
1281:
1271:
1268:
1264:
1262:
1258:
1253:
1251:
1247:
1243:
1239:
1237:
1233:
1229:
1225:
1216:
1214:
1213:Louis Mordell
1191:
1185:
1179:
1176:
1173:
1168:
1157:
1154:
1148:
1145:
1142:
1139:
1133:
1130:
1124:
1121:
1118:
1112:
1109:
1102:
1101:
1100:
1096:
1081:
1077:
1057:
1051:
1045:
1042:
1039:
1034:
1031:
1020:
1017:
1011:
1008:
1005:
1002:
996:
993:
987:
981:
978:
971:
970:
969:
961:
951:
949:
944:
939:
935:
930:
921:
916:
911:
910:
909:
907:
899:
894:
889:
888:
887:
885:
880:
878:
871:
858:
854:
850:
846:
842:
838:
835:
831:
827:
823:
819:
815:
812:
808:
804:
800:
796:
792:
788:
784:
781:
777:
773:
769:
765:
761:
758:
754:
750:
746:
742:
738:
734:
731:
727:
723:
719:
712:
705:
698:
694:
690:
686:
683:
679:
675:
671:
664:
657:
650:
646:
642:
638:
637:
636:
634:
630:
626:
622:
618:
614:
611:, giving the
610:
606:
596:
592:
584:
580:
543:
540:
537:
532:
528:
524:
521:
516:
512:
504:
503:
502:
499:
498:-smoothness.
491:
484:
470:
463:
453:
439:
408:
385:
382:
377:
373:
369:
366:
363:
358:
354:
346:
345:
344:
329:
320:
307:
305:
301:
291:The procedure
288:
258:
251:
248:
245:
242:
239:
236:
233:
227:
222:
218:
214:
207:
203:
197:
193:
189:
182:
178:
172:
168:
160:
156:
150:
146:
141:
133:
132:
131:
125:
124:prime numbers
104:
100:
96:
93:
88:
84:
80:
75:
71:
64:
61:
54:
44:
42:
38:
34:
30:
26:
22:
21:number theory
2098:
2092:
2088:
2084:
2067:
2063:
1898:
1894:
1814:
1795:(3): 42–44.
1792:
1788:
1771:
1767:
1763:
1759:
1742:
1738:
1711:
1707:
1667:
1661:
1626:
1622:
1592:(1): 83–84.
1589:
1583:
1579:
1575:
1550:
1546:
1526:
1522:
1518:
1514:
1510:
1488:
1476:
1447:
1434:
1429:, Theorem 8.
1422:
1417:, Theorem 7.
1410:
1398:
1393:, Theorem 2.
1386:
1381:, Theorem 1.
1374:
1362:
1341:minor second
1331:), 6/5 (the
1319:), 4/3 (the
1315:), 3/2 (the
1306:
1295:
1288:
1277:
1265:
1261:divisibility
1254:
1249:
1245:
1240:
1224:Chein (1976)
1222:
1211:
1094:
1079:
1075:
1072:
959:
952:
925:
905:
903:
883:
881:
873:
869:
867:
852:
848:
844:
840:
829:
825:
821:
817:
806:
798:
794:
790:
786:
775:
771:
767:
763:
756:
748:
744:
740:
736:
729:
714:
707:
700:
696:
692:
688:
681:
666:
659:
652:
648:
644:
640:
632:
628:
624:
620:
609:music theory
602:
590:
589:max(3, (max(
582:
578:
558:
500:
486:
479:
471:
458:
448:
430:
400:
327:
308:
304:D. H. Lehmer
294:
274:
50:
29:Carl Størmer
24:
18:
1553:: 257–279.
1333:minor third
1329:major third
1082:) + 1) / 2)
886:primes are
617:just tuning
593:) + 1) / 2)
426:2 − 1
412:other than
2135:Categories
1502:References
1457:Luo (1991)
1453:Cao (1991)
1337:minor tone
1325:whole step
1099:, we have
940:(sequence
938:triangular
847:− 60
824:− 30
793:− 20
770:− 12
743:− 10
53:finite set
2070:: 20–24.
1953:−
1930:−
1878:±
1859:−
1714:: 57–79.
1636:1212.5161
1248:+ 1, for
1180:
1174:−
1149:
1122:−
1113:
1046:
1040:−
1012:
982:
695:− 6
647:− 2
522:−
401:for each
364:−
325:. Assume
252:…
228:∈
215:∣
190:⋯
126:then the
97:…
47:Statement
1811:(1974).
1702:(1964).
879:+1)/2).
575:, where
447:max(3, (
405:-smooth
2123:0211954
2115:2314877
2091:= ±1".
2076:1059671
1801:1490505
1780:1319302
1751:1148835
1730:0158849
1692:0313189
1684:2317424
1653:3047912
1614:0404133
1606:2041579
1535:1138803
1298:-smooth
946:in the
943:A117583
932:in the
929:A117582
918:in the
915:A117581
896:in the
893:A002071
857:(15,16)
811:(80,81)
726:(24,25)
599:Example
454:+ 1)/2)
2121:
2113:
2074:
1827:
1799:
1778:
1766:+1)".
1749:
1728:
1690:
1682:
1651:
1612:
1604:
1582:+ 1".
1567:769515
1565:
1533:
1517:-1) =
1513:- 1)/(
1467:, and
1313:octave
1291:-limit
1236:square
1073:where
753:(9,10)
619:) let
559:where
492:+ 1)/2
319:smooth
2111:JSTOR
1680:JSTOR
1631:arXiv
1602:JSTOR
1563:JSTOR
1355:Notes
936:) or
908:, is
834:(5,6)
803:(4,5)
780:(3,4)
722:(2,3)
678:(8,9)
674:(1,2)
1984:and
1901:(2).
1825:ISBN
1131:<
1084:and
994:<
957:and
948:OEIS
934:OEIS
920:OEIS
898:OEIS
839:For
816:For
785:For
762:For
735:For
724:and
687:For
676:and
639:For
615:for
607:(in
309:Let
2103:doi
2062:".
1716:doi
1672:doi
1641:doi
1594:doi
1555:doi
1521:".
1515:abx
1177:log
1146:log
1110:log
1097:± 1
1043:log
1009:log
979:log
435:, y
333:= 2
122:of
19:In
2137::
2119:MR
2117:.
2109:.
2099:74
2097:.
2089:nY
2087:-
2085:mX
2072:MR
2068:26
2066:.
1897:.
1823:.
1821:73
1797:MR
1793:18
1776:MR
1772:63
1770:.
1747:MR
1743:28
1741:.
1726:MR
1724:.
1710:.
1706:.
1688:MR
1686:.
1678:.
1668:79
1666:.
1649:MR
1647:.
1639:.
1627:22
1625:.
1610:MR
1608:.
1600:.
1590:56
1588:.
1578:=
1561:.
1549:.
1531:MR
1527:36
1525:.
1519:by
1511:ax
1463:,
1459:,
1455:,
1238:.
962:+1
922:).
900:).
581:,
469:.
23:,
2125:.
2105::
2078:.
2048:2
2044:y
2040:=
2037:)
2034:1
2031:+
2028:x
2025:a
2022:(
2018:/
2014:)
2011:1
2008:+
2003:n
1999:x
1995:a
1992:(
1970:2
1966:y
1962:=
1959:)
1956:1
1950:x
1947:a
1944:(
1940:/
1936:)
1933:1
1925:n
1921:x
1917:a
1914:(
1899:I
1881:1
1875:=
1870:2
1866:y
1862:D
1854:2
1850:x
1833:.
1803:.
1782:.
1764:x
1762:(
1760:P
1753:.
1732:.
1718::
1712:8
1694:.
1674::
1655:.
1643::
1633::
1616:.
1596::
1580:y
1576:x
1569:.
1557::
1551:4
1537:.
1495:.
1483:.
1471:.
1442:.
1405:.
1369:.
1296:n
1289:n
1250:x
1246:x
1192:.
1189:)
1186:2
1183:(
1169:S
1164:)
1161:)
1158:S
1155:4
1152:(
1143:+
1140:2
1137:(
1134:M
1128:)
1125:1
1119:x
1116:(
1095:x
1090:P
1086:S
1080:P
1076:M
1058:,
1055:)
1052:4
1049:(
1035:S
1032:2
1027:)
1024:)
1021:S
1018:8
1015:(
1006:+
1003:2
1000:(
997:M
991:)
988:x
985:(
966:P
960:x
955:x
906:k
884:k
876:k
874:p
870:k
859:.
853:P
849:y
845:x
841:q
836:.
830:P
826:y
822:x
818:q
813:.
807:P
799:P
795:y
791:x
787:q
782:.
776:P
772:y
768:x
764:q
757:P
749:P
745:y
741:x
737:q
730:P
717:i
715:x
710:i
708:x
703:i
701:x
697:y
693:x
689:q
682:P
669:i
667:x
662:i
660:x
655:i
653:x
649:y
645:x
641:q
633:P
629:q
625:P
621:P
591:P
585:)
583:y
579:x
577:(
569:P
565:P
561:D
544:,
541:1
538:=
533:2
529:y
525:D
517:2
513:x
496:P
489:i
487:x
482:i
480:x
478:(
474:P
467:P
461:k
459:p
451:k
449:p
443:i
437:i
433:i
431:x
422:P
418:q
414:2
410:q
403:P
386:1
383:=
378:2
374:y
370:q
367:2
359:2
355:x
341:P
337:P
331:1
328:p
323:P
317:-
315:P
311:P
297:3
285:P
281:P
277:P
259:}
255:}
249:,
246:2
243:,
240:1
237:,
234:0
231:{
223:i
219:e
208:k
204:e
198:k
194:p
183:2
179:e
173:2
169:p
161:1
157:e
151:1
147:p
142:{
128:P
110:}
105:k
101:p
94:,
89:2
85:p
81:,
76:1
72:p
68:{
65:=
62:P
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