749:
601:
As each expansion step reduces the numerator of the remaining fraction to be expanded, this method always terminates with a finite expansion; however, compared to ancient
Egyptian expansions or to more modern methods, this method may produce expansions that are quite long, with large denominators.
1749:
357:
1780:
based on the greedy method. Their algorithm computes the greedy expansion of a root; at each step in this expansion it maintains an auxiliary polynomial that has as its root the remaining fraction to be expanded. Consider as an example applying this method to find the greedy expansion of the
1433:
605:
2370:
452:
138:
for a more detailed listing of these methods. As Salzer (1948) details, the greedy method, and extensions of it for the approximation of irrational numbers, have been rediscovered several times by modern mathematicians, earliest and most notably by
2597:
However, it may be difficult to determine whether an algorithm of this type can always succeed in finding a finite expansion. In particular, it is unknown whether the odd greedy expansion terminates with a finite expansion for all fractions
1075:
253:
818:
1624:
2515:
1329:
133:
Fibonacci actually lists several different methods for constructing
Egyptian fraction representations. He includes the greedy method as a last resort for situations when several simpler methods fail; see
2444:
In general, if one wants an
Egyptian fraction expansion in which the denominators are constrained in some way, it is possible to define a greedy algorithm in which at each step one chooses the expansion
362:
2284:
1001:
744:{\displaystyle {\frac {5}{121}}={\frac {1}{25}}+{\frac {1}{757}}+{\frac {1}{763\,309}}+{\frac {1}{873\,960\,180\,913}}+{\frac {1}{1\,527\,612\,795\,642\,093\,418\,846\,225}},}
754:
2564:
2624:
248:
2644:
2584:
2535:
204:
172:
2390:
The length, minimum denominator, and maximum denominator of the greedy expansion for all fractions with small numerators and denominators can be found in the
147:) A closely related expansion method that produces closer approximations at each step by allowing some unit fractions in the sum to be negative dates back to
1744:{\displaystyle 1,{\frac {2}{3}},{\frac {3}{7}},{\frac {4}{17}},{\frac {5}{31}},{\frac {6}{109}},{\frac {7}{253}},{\frac {8}{97}},{\frac {9}{271}},\dots }
2448:
352:{\displaystyle {\frac {x}{y}}={\frac {1}{\left\lceil {\frac {y}{x}}\right\rceil }}+{\frac {(-y){\bmod {x}}}{y\left\lceil {\frac {y}{x}}\right\rceil }}}
2416:
2406:
2396:
2391:
2379:
1758:
934:
3179:
3174:
3169:
2724:
Cahen, E. (1891), "Note sur un développement des quantités numériques, qui presente quelque analogie avec celui en fractions continues",
942:) can be viewed as generated by an infinite greedy expansion of this type for the number 1, where at each step we choose the denominator
3098:
2913:
3034:
1547:
839:. The greedy method leads to an expansion with ten terms, the last of which has over 500 digits in its denominator; however,
2867:
2828:
2743:
1428:{\displaystyle {\frac {(-y){\bmod {x}}}{y\left\lceil {\frac {y}{x}}\right\rceil }}={\frac {2}{\,{\frac {y(y+2)}{3}}\,}}}
126:(Fibonacci). It is called a greedy algorithm because at each step the algorithm chooses greedily the largest possible
3214:
2018:
and 1, and the first term in its greedy expansion (the second term in the greedy expansion for the golden ratio) is
3219:
3184:
2365:{\displaystyle \varphi ={\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{9}}+{\frac {1}{145}}+{\frac {1}{37986}}+\cdots }
1101:
describes an application of these closest-approximation results in lower-bounding the number of divisors of a
447:{\displaystyle {\frac {7}{15}}={\frac {1}{3}}+{\frac {2}{15}}={\frac {1}{3}}+{\frac {1}{8}}+{\frac {1}{120}}.}
3091:
2792:
929:
1777:
140:
116:, but the first published systematic method for constructing such expansions was described in 1202 in the
2646:
is odd, although it is possible to find finite odd expansions for these fractions by non-greedy methods.
2738:
1081:-term Egyptian fraction. That is, for example, any Egyptian fraction for a number in the open interval (
2591:
2586:
is distinct from all previously chosen denominators. Examples of methods defined in this way include
2279:
Continuing this approximation process eventually produces the greedy expansion for the golden ratio,
113:
55:
3000:
Stratemeyer, G. (1930), "Stammbruchentwickelungen für die
Quadratwurzel aus einer rationalen Zahl",
3190:
3130:
1884:
is the remaining fraction after the first step of the greedy expansion, it satisfies the equation
3148:
3084:
3051:
3017:
2988:
2960:
2932:
2884:
2845:
2760:
1070:{\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{43}}={\frac {1805}{1806}}}
2865:
Salzer, H. E. (1948), "Further remarks on the approximation of numbers as sums of reciprocals",
2537:
is chosen, among all possible values satisfying the constraints, as small as possible such that
2436:, though not labeled as being produced by the greedy algorithm, appear to be of the same type.
2909:
2429:
2425:
135:
51:
2540:
2206:
is the remaining fraction after this step of the greedy expansion, it satisfies the equation
2041:
is the remaining fraction after this step of the greedy expansion, it satisfies the equation
17:
3043:
3009:
2972:
2876:
2837:
2752:
211:
123:
39:
2984:
2896:
2857:
2818:
2784:
3029:
2980:
2892:
2853:
2814:
2780:
2587:
1861:) between 1 and 2. That is, the first term of the greedy expansion of the golden ratio is
454:
in this expansion, the denominator 3 of the first unit fraction is the result of rounding
47:
2601:
225:
2629:
2569:
2520:
1102:
189:
157:
3208:
3021:
2992:
2772:
127:
59:
1807:. The algorithm of Stratemeyer and Salzer performs the following sequence of steps:
2590:, in which each successive denominator must be a multiple of the previous one, and
1782:
1575:
such expansions must be found by methods other than the greedy algorithm, with the
1110:
813:{\displaystyle {\frac {5}{121}}={\frac {1}{33}}+{\frac {1}{121}}+{\frac {1}{363}}.}
2239:, which can again be expanded as a polynomial equation with integer coefficients,
2433:
210:
usually refers, not to this method, but to representation of integers as sums of
3122:
1326:
is odd, so the fraction remaining after a single step of the greedy expansion,
118:
31:
359:(simplifying the second term in this replacement as necessary). For instance:
3063:
1153:
examine the conditions under which the greedy method produces an expansion of
112:. As the name indicates, these representations have been used as long ago as
2826:
Salzer, H. E. (1947), "The approximation of numbers as sums of reciprocals",
3107:
2428:
leads to an infinite increasing sequence of integers, and the OEIS contains
518:. The denominator of the second unit fraction, 8, is the result of rounding
43:
2805:
Mays, Michael (1987), "A worst case of the
Fibonacci–Sylvester expansion",
2775:; Phillips, G. M. (1999), "Sylvester's algorithm and Fibonacci numbers",
1203:
requires one term in its greedy expansion; the simplest such fraction is
2937:
1611:-term greedy expansions and that have the smallest possible denominator
3055:
3013:
2976:
2888:
2849:
2764:
3047:
2880:
2841:
2756:
1177:
terms; these can be described in terms of congruence conditions on
130:
that can be used in any representation of the remaining fraction.
2510:{\displaystyle {\frac {x}{y}}={\frac {1}{d}}+{\frac {xd-y}{yd}},}
994:
terms and forming the corresponding
Egyptian fraction, e.g. (for
2807:
3080:
2777:
Applications of
Fibonacci numbers, Vol. 8 (Rochester, NY, 1998)
2594:, in which all denominators are constrained to be odd numbers.
1349:
314:
3076:
1284:
requires three terms in its greedy expansion if and only if
250:
to be represented, by repeatedly performing the replacement
2420:
2410:
2400:
2374:
1753:
1490:
requires four terms in its greedy expansion if and only if
938:
2946:
Spiess, O. (1907), "Über eine Klasse unendlicher Reihen",
1240:
requires two terms in its greedy expansion if and only if
1077:
results in the closest possible underestimate of 1 by any
534:
up to the next larger integer, and the remaining fraction
470:
up to the next larger integer, and the remaining fraction
2797:
Beyträge zum
Gebrauche der Mathematik und deren Anwendung
2424:, respectively. In addition, the greedy expansion of any
3032:(1880), "On a point in the theory of vulgar fractions",
2963:(1983), "Pseudofree actions and the greedy algorithm",
1785:, one of the two solutions of the polynomial equation
1568:
have an expansion with three or fewer terms, but when
1508:
of the remaining fraction is 3 and the denominator is
751:
while other methods lead to the much better expansion
2632:
2604:
2572:
2543:
2523:
2451:
2287:
1627:
1332:
1004:
757:
608:
365:
256:
228:
192:
160:
3162:
3141:
3114:
2779:, Dordrecht: Kluwer Acad. Publ., pp. 155–163,
1117:
Maximum-length expansions and congruence conditions
154:The expansion produced by this method for a number
2638:
2618:
2578:
2558:
2529:
2509:
2364:
1743:
1579:case being covered by the congruence relationship
1427:
1069:
924:
812:
743:
446:
351:
242:
198:
166:
54:. An Egyptian fraction is a representation of an
1150:
925:Sylvester's sequence and closest approximation
855:has a much shorter non-greedy representation,
3092:
2707:
823:suggests an even more badly-behaved example,
27:Simple method for finding Egyptian fractions.
8:
2741:(1922), "On Kellogg's diophantine problem",
1435:is in simplest terms. The simplest fraction
2183:, the next term in the greedy expansion is
1778:approximation for the roots of a polynomial
1769:
222:Fibonacci's algorithm expands the fraction
3099:
3085:
3077:
2430:expansions of several well known constants
2936:
2631:
2608:
2603:
2571:
2542:
2522:
2478:
2465:
2452:
2450:
2392:On-Line Encyclopedia of Integer Sequences
2346:
2333:
2320:
2307:
2294:
2286:
1776:describe a method of finding an accurate
1725:
1712:
1699:
1686:
1673:
1660:
1647:
1634:
1626:
1587:More generally the sequence of fractions
1421:
1394:
1393:
1387:
1367:
1352:
1348:
1333:
1331:
1057:
1044:
1031:
1018:
1005:
1003:
797:
784:
771:
758:
756:
731:
727:
723:
719:
715:
711:
707:
703:
694:
684:
680:
676:
667:
657:
648:
635:
622:
609:
607:
431:
418:
405:
392:
379:
366:
364:
332:
317:
313:
298:
279:
270:
257:
255:
232:
227:
191:
159:
144:
2799:, Berlin: Zweyter Theil, pp. 99–104
1097:, 1) requires at least five terms.
3154:Greedy algorithm for Egyptian fractions
2703:
2655:
1098:
148:
36:greedy algorithm for Egyptian fractions
2691:
2674:
2662:
1773:
2904:Sigler, Laurence E. (trans.) (2002),
2687:
1106:
820:
7:
3180:Generalizations of Fibonacci numbers
3175:List of things named after Fibonacci
3170:Fibonacci numbers in popular culture
1146:
2726:Nouvelles Annales des Mathématiques
602:For instance, this method expands
25:
3070:, W. H. Freeman, pp. 271–277
2925:Approximating 1 from below using
1765:Approximation of polynomial roots
2948:Archiv der Mathematik und Physik
1247:; the simplest such fraction is
3035:American Journal of Mathematics
1453:with a three-term expansion is
1145:terms in its greedy expansion.
2434:additional entries in the OEIS
1530:with a four-term expansion is
1412:
1400:
1345:
1336:
990:. Truncating this sequence to
310:
301:
1:
2868:American Mathematical Monthly
2829:American Mathematical Monthly
2744:American Mathematical Monthly
1151:Freitag & Phillips (1999)
486:is the result of simplifying
184:Fibonacci–Sylvester expansion
18:Fibonacci–Sylvester expansion
2074:, which can be expanded as
1901:, which can be expanded as
3236:
2923:Soundararajan, K. (2005),
1850:, there must be a root of
1550:states that all fractions
1109:describes applications in
3185:The Fibonacci Association
3002:Mathematische Zeitschrift
1497:, for then the numerator
176:greedy Egyptian expansion
1512:. The simplest fraction
932:2, 3, 7, 43, 1807, ... (
2906:Fibonacci's Liber Abaci
2559:{\displaystyle xd>y}
2386:Other integer sequences
1548:Erdős–Straus conjecture
566:after subtracting both
2640:
2620:
2580:
2560:
2531:
2511:
2366:
1745:
1429:
1071:
814:
745:
448:
353:
244:
218:Algorithm and examples
200:
168:
3068:Mathematica in Action
2965:Mathematische Annalen
2641:
2621:
2581:
2561:
2532:
2512:
2367:
1746:
1430:
1072:
815:
746:
550:is what is left from
449:
354:
245:
201:
169:
58:as a sum of distinct
42:, first described by
2630:
2602:
2592:odd greedy expansion
2570:
2541:
2521:
2449:
2285:
1829: = 1, and
1625:
1330:
1002:
930:Sylvester's sequence
755:
606:
363:
254:
226:
206:. However, the term
190:
158:
56:irreducible fraction
3191:Fibonacci Quarterly
3131:The Book of Squares
2908:, Springer-Verlag,
2619:{\displaystyle x/y}
243:{\displaystyle x/y}
208:Fibonacci expansion
180:Sylvester expansion
141:J. J. Sylvester
46:, for transforming
3215:Egyptian fractions
3149:Fibonacci sequence
3014:10.1007/BF01246446
2977:10.1007/BF01455950
2929:Egyptian fractions
2708:Soundararajan 2005
2636:
2616:
2576:
2556:
2527:
2507:
2440:Related expansions
2362:
1770:Stratemeyer (1930)
1741:
1573:≡ 1 or 17 (mod 24)
1495:≡ 1 or 17 (mod 24)
1425:
1067:
810:
741:
444:
349:
240:
196:
164:
52:Egyptian fractions
3220:Greedy algorithms
3202:
3201:
2686:See for instance
2639:{\displaystyle y}
2579:{\displaystyle d}
2530:{\displaystyle d}
2502:
2473:
2460:
2426:irrational number
2354:
2341:
2328:
2315:
2302:
1733:
1720:
1707:
1694:
1681:
1668:
1655:
1642:
1423:
1419:
1382:
1375:
1141:requires at most
1065:
1052:
1039:
1026:
1013:
805:
792:
779:
766:
736:
689:
662:
643:
630:
617:
439:
426:
413:
400:
387:
374:
347:
340:
293:
287:
265:
212:Fibonacci numbers
199:{\displaystyle x}
167:{\displaystyle x}
136:Egyptian fraction
16:(Redirected from
3227:
3101:
3094:
3087:
3078:
3071:
3058:
3030:Sylvester, J. J.
3024:
2995:
2955:
2950:, Third Series,
2941:
2940:
2918:
2899:
2860:
2821:
2800:
2787:
2767:
2733:
2710:
2701:
2695:
2684:
2678:
2672:
2666:
2660:
2645:
2643:
2642:
2637:
2625:
2623:
2622:
2617:
2612:
2585:
2583:
2582:
2577:
2565:
2563:
2562:
2557:
2536:
2534:
2533:
2528:
2516:
2514:
2513:
2508:
2503:
2501:
2493:
2479:
2474:
2466:
2461:
2453:
2423:
2413:
2403:
2377:
2371:
2369:
2368:
2363:
2355:
2347:
2342:
2334:
2329:
2321:
2316:
2308:
2303:
2295:
2274:
2265:
2264:
2238:
2236:
2234:
2233:
2230:
2227:
2198:
2196:
2195:
2192:
2189:
2182:
2181:
2179:
2178:
2175:
2172:
2160:
2146:
2144:
2143:
2140:
2137:
2126:
2109:
2100:
2099:
2073:
2071:
2069:
2068:
2065:
2062:
2033:
2031:
2030:
2027:
2024:
2017:
2015:
2014:
2011:
2008:
1994:
1987:
1973:
1971:
1970:
1967:
1964:
1953:
1936:
1927:
1926:
1900:
1876:
1874:
1873:
1870:
1867:
1849:
1842:
1824:
1806:
1756:
1750:
1748:
1747:
1742:
1734:
1726:
1721:
1713:
1708:
1700:
1695:
1687:
1682:
1674:
1669:
1661:
1656:
1648:
1643:
1635:
1606:
1604:
1603:
1598:
1595:
1582:
1578:
1574:
1567:
1565:
1564:
1559:
1556:
1545:
1543:
1542:
1539:
1536:
1529:
1527:
1526:
1521:
1518:
1511:
1507:
1496:
1489:
1487:
1486:
1481:
1478:
1468:
1466:
1465:
1462:
1459:
1452:
1450:
1449:
1444:
1441:
1434:
1432:
1431:
1426:
1424:
1422:
1420:
1415:
1395:
1388:
1383:
1381:
1380:
1376:
1368:
1358:
1357:
1356:
1334:
1325:
1323:
1322:
1319:
1316:
1302:
1290:
1283:
1281:
1280:
1275:
1272:
1262:
1260:
1259:
1256:
1253:
1246:
1239:
1237:
1236:
1231:
1228:
1218:
1216:
1215:
1212:
1209:
1202:
1200:
1199:
1194:
1191:
1172:
1170:
1169:
1164:
1161:
1140:
1138:
1137:
1132:
1129:
1096:
1094:
1093:
1090:
1087:
1076:
1074:
1073:
1068:
1066:
1058:
1053:
1045:
1040:
1032:
1027:
1019:
1014:
1006:
998: = 4)
989:
987:
985:
984:
979:
976:
965:
963:
961:
960:
955:
952:
941:
920:
919:
917:
916:
913:
910:
903:
901:
900:
897:
894:
887:
885:
884:
881:
878:
871:
869:
868:
865:
862:
854:
852:
851:
848:
845:
838:
836:
835:
832:
829:
819:
817:
816:
811:
806:
798:
793:
785:
780:
772:
767:
759:
750:
748:
747:
742:
737:
735:
695:
690:
688:
668:
663:
661:
649:
644:
636:
631:
623:
618:
610:
597:
595:
594:
591:
588:
581:
579:
578:
575:
572:
565:
563:
562:
559:
556:
549:
547:
546:
543:
540:
533:
531:
530:
527:
524:
517:
515:
514:
511:
508:
501:
499:
498:
495:
492:
485:
483:
482:
479:
476:
469:
467:
466:
463:
460:
453:
451:
450:
445:
440:
432:
427:
419:
414:
406:
401:
393:
388:
380:
375:
367:
358:
356:
355:
350:
348:
346:
345:
341:
333:
323:
322:
321:
299:
294:
292:
288:
280:
271:
266:
258:
249:
247:
246:
241:
236:
205:
203:
202:
197:
173:
171:
170:
165:
124:Leonardo of Pisa
111:
110:
108:
107:
104:
101:
94:
92:
91:
88:
85:
78:
76:
75:
72:
69:
48:rational numbers
40:greedy algorithm
21:
3235:
3234:
3230:
3229:
3228:
3226:
3225:
3224:
3205:
3204:
3203:
3198:
3158:
3137:
3110:
3105:
3075:
3062:
3048:10.2307/2369261
3028:
2999:
2959:
2945:
2938:math.CA/0502247
2922:
2916:
2903:
2881:10.2307/2304960
2864:
2842:10.2307/2305906
2825:
2804:
2791:
2771:
2757:10.2307/2299023
2751:(10): 380–387,
2737:
2723:
2719:
2714:
2713:
2702:
2698:
2685:
2681:
2673:
2669:
2661:
2657:
2652:
2628:
2627:
2600:
2599:
2588:Engel expansion
2568:
2567:
2539:
2538:
2519:
2518:
2494:
2480:
2447:
2446:
2442:
2415:
2405:
2395:
2388:
2383:
2373:
2283:
2282:
2272:
2263:
2260:
2259:
2258:
2253:
2246:
2240:
2231:
2228:
2225:
2224:
2222:
2220:
2213:
2207:
2205:
2193:
2190:
2187:
2186:
2184:
2176:
2173:
2170:
2169:
2167:
2162:
2154:
2148:
2141:
2138:
2135:
2134:
2132:
2120:
2114:
2107:
2098:
2095:
2094:
2093:
2088:
2081:
2075:
2066:
2063:
2060:
2059:
2057:
2055:
2048:
2042:
2040:
2028:
2025:
2022:
2021:
2019:
2012:
2009:
2006:
2005:
2003:
2001:
1989:
1981:
1975:
1968:
1965:
1962:
1961:
1959:
1947:
1941:
1934:
1925:
1922:
1921:
1920:
1915:
1908:
1902:
1898:
1891:
1885:
1883:
1871:
1868:
1865:
1864:
1862:
1856:
1844:
1836:
1830:
1818:
1812:
1792:
1786:
1767:
1762:
1752:
1623:
1622:
1599:
1596:
1591:
1590:
1588:
1580:
1576:
1569:
1560:
1557:
1554:
1553:
1551:
1540:
1537:
1534:
1533:
1531:
1522:
1519:
1516:
1515:
1513:
1509:
1498:
1491:
1482:
1479:
1476:
1475:
1473:
1463:
1460:
1457:
1456:
1454:
1445:
1442:
1439:
1438:
1436:
1396:
1392:
1363:
1359:
1335:
1328:
1327:
1320:
1317:
1307:
1306:
1304:
1292:
1285:
1276:
1273:
1270:
1269:
1267:
1257:
1254:
1251:
1250:
1248:
1241:
1232:
1229:
1226:
1225:
1223:
1222:Every fraction
1213:
1210:
1207:
1206:
1204:
1195:
1192:
1189:
1188:
1186:
1185:Every fraction
1165:
1162:
1157:
1156:
1154:
1133:
1130:
1125:
1124:
1122:
1119:
1091:
1088:
1085:
1084:
1082:
1000:
999:
980:
977:
972:
971:
969:
967:
956:
953:
948:
947:
945:
943:
933:
927:
914:
911:
908:
907:
905:
898:
895:
892:
891:
889:
882:
879:
876:
875:
873:
866:
863:
860:
859:
857:
856:
849:
846:
843:
842:
840:
833:
830:
827:
826:
824:
753:
752:
699:
672:
653:
604:
603:
592:
589:
586:
585:
583:
576:
573:
570:
569:
567:
560:
557:
554:
553:
551:
544:
541:
538:
537:
535:
528:
525:
522:
521:
519:
512:
509:
506:
505:
503:
496:
493:
490:
489:
487:
480:
477:
474:
473:
471:
464:
461:
458:
457:
455:
361:
360:
328:
324:
300:
275:
252:
251:
224:
223:
220:
188:
187:
156:
155:
105:
102:
99:
98:
96:
89:
86:
83:
82:
80:
73:
70:
67:
66:
64:
63:
28:
23:
22:
15:
12:
11:
5:
3233:
3231:
3223:
3222:
3217:
3207:
3206:
3200:
3199:
3197:
3196:
3195:
3194:
3182:
3177:
3172:
3166:
3164:
3160:
3159:
3157:
3156:
3151:
3145:
3143:
3139:
3138:
3136:
3135:
3127:
3118:
3116:
3112:
3111:
3106:
3104:
3103:
3096:
3089:
3081:
3074:
3073:
3060:
3042:(4): 332–335,
3026:
2997:
2971:(4): 501–512,
2957:
2943:
2920:
2914:
2901:
2875:(6): 350–356,
2862:
2836:(3): 135–142,
2823:
2802:
2793:Lambert, J. H.
2789:
2773:Freitag, H. T.
2769:
2739:Curtiss, D. R.
2735:
2720:
2718:
2715:
2712:
2711:
2696:
2679:
2677:, chapter II.7
2667:
2654:
2653:
2651:
2648:
2635:
2615:
2611:
2607:
2575:
2566:and such that
2555:
2552:
2549:
2546:
2526:
2506:
2500:
2497:
2492:
2489:
2486:
2483:
2477:
2472:
2469:
2464:
2459:
2456:
2441:
2438:
2387:
2384:
2361:
2358:
2353:
2350:
2345:
2340:
2337:
2332:
2327:
2324:
2319:
2314:
2311:
2306:
2301:
2298:
2293:
2290:
2281:
2277:
2276:
2270:
2261:
2251:
2244:
2218:
2211:
2203:
2152:
2118:
2111:
2105:
2096:
2086:
2079:
2053:
2046:
2038:
1999:
1995:, the root of
1979:
1945:
1938:
1932:
1923:
1913:
1906:
1896:
1889:
1881:
1854:
1834:
1816:
1790:
1766:
1763:
1740:
1737:
1732:
1729:
1724:
1719:
1716:
1711:
1706:
1703:
1698:
1693:
1690:
1685:
1680:
1677:
1672:
1667:
1664:
1659:
1654:
1651:
1646:
1641:
1638:
1633:
1630:
1621:
1585:
1584:
1470:
1418:
1414:
1411:
1408:
1405:
1402:
1399:
1391:
1386:
1379:
1374:
1371:
1366:
1362:
1355:
1351:
1347:
1344:
1341:
1338:
1264:
1220:
1118:
1115:
1103:perfect number
1099:Curtiss (1922)
1064:
1061:
1056:
1051:
1048:
1043:
1038:
1035:
1030:
1025:
1022:
1017:
1012:
1009:
926:
923:
809:
804:
801:
796:
791:
788:
783:
778:
775:
770:
765:
762:
740:
734:
730:
726:
722:
718:
714:
710:
706:
702:
698:
693:
687:
683:
679:
675:
671:
666:
660:
656:
652:
647:
642:
639:
634:
629:
626:
621:
616:
613:
443:
438:
435:
430:
425:
422:
417:
412:
409:
404:
399:
396:
391:
386:
383:
378:
373:
370:
344:
339:
336:
331:
327:
320:
316:
312:
309:
306:
303:
297:
291:
286:
283:
278:
274:
269:
264:
261:
239:
235:
231:
219:
216:
195:
174:is called the
163:
149:Lambert (1770)
60:unit fractions
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3232:
3221:
3218:
3216:
3213:
3212:
3210:
3193:
3192:
3188:
3187:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3167:
3165:
3161:
3155:
3152:
3150:
3147:
3146:
3144:
3140:
3133:
3132:
3128:
3125:
3124:
3120:
3119:
3117:
3113:
3109:
3102:
3097:
3095:
3090:
3088:
3083:
3082:
3079:
3069:
3065:
3061:
3057:
3053:
3049:
3045:
3041:
3037:
3036:
3031:
3027:
3023:
3019:
3015:
3011:
3007:
3003:
2998:
2994:
2990:
2986:
2982:
2978:
2974:
2970:
2966:
2962:
2958:
2953:
2949:
2944:
2939:
2934:
2930:
2926:
2921:
2917:
2915:0-387-95419-8
2911:
2907:
2902:
2898:
2894:
2890:
2886:
2882:
2878:
2874:
2870:
2869:
2863:
2859:
2855:
2851:
2847:
2843:
2839:
2835:
2831:
2830:
2824:
2820:
2816:
2812:
2808:
2803:
2798:
2794:
2790:
2786:
2782:
2778:
2774:
2770:
2766:
2762:
2758:
2754:
2750:
2746:
2745:
2740:
2736:
2731:
2727:
2722:
2721:
2716:
2709:
2705:
2700:
2697:
2693:
2692:Spiess (1907)
2689:
2683:
2680:
2676:
2671:
2668:
2664:
2659:
2656:
2649:
2647:
2633:
2613:
2609:
2605:
2595:
2593:
2589:
2573:
2553:
2550:
2547:
2544:
2524:
2504:
2498:
2495:
2490:
2487:
2484:
2481:
2475:
2470:
2467:
2462:
2457:
2454:
2439:
2437:
2435:
2431:
2427:
2422:
2418:
2412:
2408:
2402:
2398:
2394:as sequences
2393:
2385:
2381:
2376:
2359:
2356:
2351:
2348:
2343:
2338:
2335:
2330:
2325:
2322:
2317:
2312:
2309:
2304:
2299:
2296:
2291:
2288:
2280:
2269:
2257:
2250:
2243:
2217:
2210:
2202:
2165:
2158:
2151:
2131: =
2130:
2124:
2117:
2112:
2104:
2092:
2085:
2078:
2052:
2045:
2037:
2002:lies between
1998:
1992:
1985:
1978:
1958: =
1957:
1951:
1944:
1939:
1931:
1919:
1912:
1905:
1895:
1888:
1880:
1860:
1853:
1847:
1840:
1833:
1828:
1822:
1815:
1810:
1809:
1808:
1804:
1800:
1796:
1789:
1784:
1779:
1775:
1774:Salzer (1947)
1771:
1764:
1760:
1755:
1738:
1735:
1730:
1727:
1722:
1717:
1714:
1709:
1704:
1701:
1696:
1691:
1688:
1683:
1678:
1675:
1670:
1665:
1662:
1657:
1652:
1649:
1644:
1639:
1636:
1631:
1628:
1620:
1618:
1614:
1610:
1602:
1594:
1572:
1563:
1549:
1525:
1506:
1502:
1494:
1485:
1471:
1448:
1416:
1409:
1406:
1403:
1397:
1389:
1384:
1377:
1372:
1369:
1364:
1360:
1353:
1342:
1339:
1314:
1310:
1300:
1296:
1288:
1279:
1265:
1244:
1235:
1221:
1198:
1184:
1183:
1182:
1180:
1176:
1173:with exactly
1168:
1160:
1152:
1148:
1144:
1136:
1128:
1121:Any fraction
1116:
1114:
1112:
1108:
1104:
1100:
1080:
1062:
1059:
1054:
1049:
1046:
1041:
1036:
1033:
1028:
1023:
1020:
1015:
1010:
1007:
997:
993:
983:
975:
959:
951:
940:
936:
931:
922:
822:
807:
802:
799:
794:
789:
786:
781:
776:
773:
768:
763:
760:
738:
732:
728:
724:
720:
716:
712:
708:
704:
700:
696:
691:
685:
681:
677:
673:
669:
664:
658:
654:
650:
645:
640:
637:
632:
627:
624:
619:
614:
611:
599:
502: =
441:
436:
433:
428:
423:
420:
415:
410:
407:
402:
397:
394:
389:
384:
381:
376:
371:
368:
342:
337:
334:
329:
325:
318:
307:
304:
295:
289:
284:
281:
276:
272:
267:
262:
259:
237:
233:
229:
217:
215:
213:
209:
193:
185:
181:
177:
161:
152:
150:
146:
142:
137:
131:
129:
128:unit fraction
125:
121:
120:
115:
114:ancient Egypt
61:
57:
53:
49:
45:
41:
37:
33:
19:
3189:
3153:
3129:
3121:
3067:
3039:
3033:
3005:
3001:
2968:
2964:
2961:Stong, R. E.
2951:
2947:
2928:
2924:
2905:
2872:
2866:
2833:
2827:
2810:
2806:
2796:
2776:
2748:
2742:
2729:
2725:
2704:Curtiss 1922
2699:
2688:Cahen (1891)
2682:
2670:
2658:
2596:
2443:
2389:
2278:
2267:
2255:
2248:
2241:
2215:
2208:
2200:
2163:
2156:
2149:
2128:
2122:
2115:
2102:
2090:
2083:
2076:
2050:
2043:
2035:
1996:
1990:
1983:
1976:
1955:
1949:
1942:
1929:
1917:
1910:
1903:
1893:
1886:
1878:
1858:
1851:
1845:
1838:
1831:
1826:
1820:
1813:
1802:
1798:
1794:
1787:
1783:golden ratio
1768:
1616:
1612:
1608:
1600:
1592:
1586:
1570:
1561:
1523:
1504:
1500:
1492:
1483:
1446:
1312:
1308:
1298:
1294:
1286:
1277:
1242:
1233:
1196:
1178:
1174:
1166:
1158:
1142:
1134:
1126:
1120:
1111:group theory
1107:Stong (1983)
1078:
995:
991:
981:
973:
957:
949:
928:
821:Wagon (1991)
600:
221:
207:
183:
179:
175:
153:
132:
117:
35:
29:
3123:Liber Abaci
3008:: 767–768,
2813:: 141–148,
2675:Sigler 2002
2663:Sigler 2002
1577:17 (mod 24)
1472:A fraction
1291:, for then
1289:≡ 1 (mod 6)
1266:A fraction
1245:≡ 1 (mod 2)
1147:Mays (1987)
966:instead of
119:Liber Abaci
32:mathematics
3209:Categories
2728:, Ser. 3,
2717:References
2626:for which
2372:(sequence
1751:(sequence
1607:that have
62:, such as
3108:Fibonacci
3064:Wagon, S.
3022:120956180
2993:120347233
2954:: 124–134
2732:: 508–514
2488:−
2360:⋯
2289:φ
1739:…
1615:for each
1581:2 (mod 3)
1510:1 (mod 6)
1340:−
491:−15 mod 7
305:−
44:Fibonacci
3142:Theories
3066:(1991),
2795:(1770),
2161:for all
2159:) > 0
2125:) < 0
1988:for all
1986:) > 0
1952:) < 0
1899:+ 1) = 0
1843:for all
1841:) > 0
1823:) < 0
1378:⌉
1365:⌈
1105:, while
343:⌉
330:⌈
290:⌉
277:⌈
3163:Related
3056:2369261
2985:0721884
2897:0025512
2889:2304960
2858:0020339
2850:2305906
2819:0888838
2785:1737669
2765:2299023
2432:. Some
2421:A050210
2419::
2411:A050206
2409::
2401:A050205
2399::
2378:in the
2375:A117116
2273:− 5 = 0
2254:) = 324
2235:
2223:
2197:
2185:
2180:
2168:
2145:
2133:
2108:− 1 = 0
2070:
2058:
2032:
2020:
2016:
2004:
1972:
1960:
1935:− 1 = 0
1875:
1863:
1805:− 1 = 0
1757:in the
1754:A048860
1605:
1589:
1566:
1552:
1544:
1532:
1528:
1514:
1488:
1474:
1467:
1455:
1451:
1437:
1324:
1305:
1282:
1268:
1261:
1249:
1238:
1224:
1217:
1205:
1201:
1187:
1171:
1155:
1139:
1123:
1095:
1083:
986:
970:
962:
946:
939:A000058
937::
918:
906:
902:
890:
886:
874:
870:
858:
853:
841:
837:
825:
596:
584:
580:
568:
564:
552:
548:
536:
532:
520:
516:
504:
500:
488:
484:
472:
468:
456:
143: (
109:
97:
93:
81:
77:
65:
3134:(1225)
3126:(1202)
3054:
3020:
2991:
2983:
2912:
2895:
2887:
2856:
2848:
2817:
2783:
2763:
2517:where
2414:, and
2147:, and
2113:Since
1993:> 1
1974:, and
1940:Since
1811:Since
1546:. The
497:15 × 3
34:, the
3115:Books
3052:JSTOR
3018:S2CID
2989:S2CID
2933:arXiv
2885:JSTOR
2846:JSTOR
2761:JSTOR
2650:Notes
2352:37986
2266:+ 720
2237:) = 0
2199:. If
2166:>
2089:) = 4
2072:) = 0
2034:. If
1877:. If
964:⌋ + 1
182:, or
50:into
38:is a
2910:ISBN
2690:and
2551:>
2417:OEIS
2407:OEIS
2397:OEIS
2380:OEIS
2127:for
1954:for
1916:) =
1825:for
1797:) =
1772:and
1759:OEIS
1503:mod
1315:+ 2)
1303:and
1297:mod
1149:and
1092:1806
1086:1805
1063:1806
1060:1805
935:OEIS
915:8708
899:2799
582:and
145:1880
3044:doi
3010:doi
2973:doi
2969:265
2877:doi
2838:doi
2753:doi
2339:145
2101:+ 8
1848:≥ 2
1731:271
1705:253
1692:109
1619:is
1350:mod
1301:= 2
850:311
834:311
803:363
790:121
764:121
733:225
729:846
725:418
721:093
717:642
713:795
709:612
705:527
686:913
682:180
678:960
674:873
659:309
655:763
641:757
615:121
545:120
437:120
315:mod
186:of
122:of
30:In
3211::
3050:,
3038:,
3016:,
3006:31
3004:,
2987:,
2981:MR
2979:,
2967:,
2952:12
2931:,
2893:MR
2891:,
2883:,
2873:55
2871:,
2854:MR
2852:,
2844:,
2834:54
2832:,
2815:MR
2809:,
2781:MR
2759:,
2749:29
2747:,
2730:10
2706:;
2404:,
2382:).
2221:+
2056:+
1928:+
1801:−
1761:).
1718:97
1679:31
1666:17
1541:17
1181:.
1113:.
1050:43
968:⌈
944:⌊
921:.
904:+
888:+
883:63
872:+
867:12
844:31
828:31
777:33
628:25
598:.
561:15
523:15
513:45
481:15
459:15
398:15
372:15
214:.
178:,
151:.
95:+
79:=
3100:e
3093:t
3086:v
3072:.
3059:.
3046::
3040:3
3025:.
3012::
2996:.
2975::
2956:.
2942:.
2935::
2927:n
2919:.
2900:.
2879::
2861:.
2840::
2822:.
2811:1
2801:.
2788:.
2768:.
2755::
2734:.
2694:.
2665:.
2634:y
2614:y
2610:/
2606:x
2574:d
2554:y
2548:d
2545:x
2525:d
2505:,
2499:d
2496:y
2491:y
2485:d
2482:x
2476:+
2471:d
2468:1
2463:=
2458:y
2455:x
2357:+
2349:1
2344:+
2336:1
2331:+
2326:9
2323:1
2318:+
2313:2
2310:1
2305:+
2300:1
2297:1
2292:=
2275:.
2271:3
2268:x
2262:3
2256:x
2252:3
2249:x
2247:(
2245:3
2242:P
2232:9
2229:/
2226:1
2219:3
2216:x
2214:(
2212:2
2209:P
2204:3
2201:x
2194:9
2191:/
2188:1
2177:8
2174:/
2171:1
2164:x
2157:x
2155:(
2153:2
2150:P
2142:9
2139:/
2136:1
2129:x
2123:x
2121:(
2119:2
2116:P
2110:.
2106:2
2103:x
2097:2
2091:x
2087:2
2084:x
2082:(
2080:2
2077:P
2067:2
2064:/
2061:1
2054:2
2051:x
2049:(
2047:1
2044:P
2039:2
2036:x
2029:2
2026:/
2023:1
2013:2
2010:/
2007:1
2000:1
1997:P
1991:x
1984:x
1982:(
1980:1
1977:P
1969:2
1966:/
1963:1
1956:x
1950:x
1948:(
1946:1
1943:P
1937:.
1933:1
1930:x
1924:1
1918:x
1914:1
1911:x
1909:(
1907:1
1904:P
1897:1
1894:x
1892:(
1890:0
1887:P
1882:1
1879:x
1872:1
1869:/
1866:1
1859:x
1857:(
1855:0
1852:P
1846:x
1839:x
1837:(
1835:0
1832:P
1827:x
1821:x
1819:(
1817:0
1814:P
1803:x
1799:x
1795:x
1793:(
1791:0
1788:P
1736:,
1728:9
1723:,
1715:8
1710:,
1702:7
1697:,
1689:6
1684:,
1676:5
1671:,
1663:4
1658:,
1653:7
1650:3
1645:,
1640:3
1637:2
1632:,
1629:1
1617:x
1613:y
1609:x
1601:y
1597:/
1593:x
1583:.
1571:y
1562:y
1558:/
1555:4
1538:/
1535:4
1524:y
1520:/
1517:4
1505:x
1501:y
1499:−
1493:y
1484:y
1480:/
1477:4
1469:.
1464:7
1461:/
1458:3
1447:y
1443:/
1440:3
1417:3
1413:)
1410:2
1407:+
1404:y
1401:(
1398:y
1390:2
1385:=
1373:x
1370:y
1361:y
1354:x
1346:)
1343:y
1337:(
1321:3
1318:/
1313:y
1311:(
1309:y
1299:x
1295:y
1293:−
1287:y
1278:y
1274:/
1271:3
1263:.
1258:3
1255:/
1252:2
1243:y
1234:y
1230:/
1227:2
1219:.
1214:1
1211:/
1208:1
1197:y
1193:/
1190:1
1179:y
1175:x
1167:y
1163:/
1159:x
1143:x
1135:y
1131:/
1127:x
1089:/
1079:k
1055:=
1047:1
1042:+
1037:7
1034:1
1029:+
1024:3
1021:1
1016:+
1011:2
1008:1
996:k
992:k
988:⌉
982:x
978:/
974:y
958:x
954:/
950:y
912:/
909:1
896:/
893:1
880:/
877:1
864:/
861:1
847:/
831:/
808:.
800:1
795:+
787:1
782:+
774:1
769:=
761:5
739:,
701:1
697:1
692:+
670:1
665:+
651:1
646:+
638:1
633:+
625:1
620:=
612:5
593:8
590:/
587:1
577:3
574:/
571:1
558:/
555:7
542:/
539:1
529:2
526:/
510:/
507:6
494:/
478:/
475:2
465:7
462:/
442:.
434:1
429:+
424:8
421:1
416:+
411:3
408:1
403:=
395:2
390:+
385:3
382:1
377:=
369:7
338:x
335:y
326:y
319:x
311:)
308:y
302:(
296:+
285:x
282:y
273:1
268:=
263:y
260:x
238:y
234:/
230:x
194:x
162:x
106:3
103:/
100:1
90:2
87:/
84:1
74:6
71:/
68:5
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.