7594:
1065:
323, 35, 119, 9, 9, 143, 25, 33, 9, 15, 123, 35, 9, 9, 15, 129, 51, 9, 33, 15, 21, 9, 9, 49, 15, 39, 9, 35, 49, 15, 9, 9, 33, 51, 15, 9, 35, 85, 39, 9, 9, 21, 25, 51, 9, 143, 33, 119, 9, 9, 51, 33, 95, 9, 15, 301, 25, 9, 9, 15, 49, 155, 9, 399, 15, 33, 9, 9, 49, 15, 119, 9,
4297:
1750:
4913:
2096:
4102:
2695:
255:
4694:
pseudoprimes. Hoggatt and
Bicknell studied properties of these pseudoprimes in 1974. Singmaster computed these pseudoprimes up to 100000. Jacobsen lists all 111443 of these pseudoprimes less than 10.
1131:
2885:
890:
1818:
1323:
336:
2790:
559:
3589:
3492:
2025:
1956:
945:
4624:
2147:
818:
1905:, the number to be tested for primality, is odd, is not a perfect square, and is not divisible by any small prime less than some convenient limit. Perfect squares are easy to detect using
1000:
4143:
3786:
3655:
499:
187:
628:
1255:
2582:
2999:
2941:
3673:
Although this congruence condition is not, by definition, part of the Lucas probable prime test, it is almost free to check this condition because, as noted above, the value of
5696:
4168:
1859:
2397:
1539:
1164:
4002:
3866:
3446:
109:
2477:
2231:
3035:
1427:
3935:
3902:
2437:
2355:
2322:
1891:
1620:
1593:
3400:
2185:
2523:
3955:
3323:
are chosen as described above, the first 10 Lucas pseudoprimes are (see page 1401 of ): 323, 377, 1159, 1829, 3827, 5459, 5777, 9071, 9179, and 10877 (sequence
3121:
3101:
3081:
2500:
1559:
1507:
1487:
1467:
1447:
1391:
1371:
1184:
509:. Typically implementations will use a parameter selection method that ensures this condition (e.g. the Selfridge method recommended in and described below).
1666:
4833:
4704:
4683:
4517:
3517:
3366:
3330:
740:
5689:
4847:
4697:
It has been shown that there are no even
Fibonacci pseudoprimes as defined by equation (5). However, even Fibonacci pseudoprimes do exist (sequence
2030:
4539:
is congruent to 1 or 4 modulo 5, the opposite is true, with over 12% of
Fibonacci pseudoprimes under 10 also being base-2 Fermat pseudoprimes.
4021:
2588:
4531:
is congruent to 2 or 3 modulo 5, then
Bressoud, and Crandall and Pomerance point out that it is rare for a Fibonacci pseudoprime to also be a
5075:
501:
is −1 (see pages 1401â1409 of, page 1024 of, or pages 266â269 of ). This is especially important when combining a Lucas test with a
4330:
6496:
5682:
5553:
MĂŒller, Winfried B.; Oswald, Alan (1993). "Generalized
Fibonacci Pseudoprimes and Probable Primes". In G.E. Bergum; et al. (eds.).
6491:
6506:
6486:
3146:+1 = 44 (= 101100 in binary), then, taking the bits one at a time from left to right, we obtain the sequence of indices to compute: 1
5153:
5116:
4315:
The Jacobi symbol on the right side is easy to compute, so this congruence is easy to check. If this congruence does not hold, then
1353:
There are infinitely many strong Lucas pseudoprimes, and therefore, infinitely many Lucas pseudoprimes. Theorem 7 in states: Let
7199:
6779:
2157:
values that must be tried before we encounter one whose Jacobi symbol is −1 is about 1.79; see, p. 1416. Once we have
4360:
199:
1078:
4447:
2796:
506:
6501:
7285:
837:
699:
is prime. As is the case with any other probabilistic primality test, if we perform additional Lucas tests with different
395:. Hence, every Frobenius pseudoprime is also a Baillie-Wagstaff-Lucas pseudoprime, but the converse does not always hold.
6601:
3534:) is true, there are additional congruence conditions we can check that have almost no additional computational cost. If
1761:
1266:
6951:
6270:
6063:
4948:) = (2, â1), leading to the pseudoprimes 169, 385, 961, 1105, 1121, 3827, 4901, 6265, 6441, 6601, 7107, 7801, 8119, ...
280:
6986:
6956:
6631:
6621:
2707:
7127:
6541:
6275:
6255:
515:
6817:
3548:
3451:
1984:
1915:
913:
6981:
4562:
2109:
786:
7618:
7076:
6699:
6456:
6265:
6247:
6141:
6131:
6121:
5595:(data for Lucas, Strong Lucas, AES Lucas, ES Lucas pseudoprimes below 10; Fibonacci and Pell pseudoprimes below 10)
953:
6961:
4111:
3754:
3599:
467:
155:
7204:
6749:
6370:
6156:
6151:
6146:
6136:
6113:
575:
6189:
3818:) is true (see page 1409 and Table 6 of;). More extensive calculations show that, with this method of choosing
6446:
1211:
2531:
7315:
7280:
7066:
6976:
6850:
6825:
6734:
6724:
6336:
6318:
6238:
5424:
5284:
5177:
4993:
2946:
3868:, then a further congruence condition that involves very little additional computation can be implemented.
7575:
6845:
6719:
6350:
6126:
5906:
5833:
5530:
5404:
5395:
Di Porto, Adina; Filipponi, Piero; Montolivo, Emilio (1990). "On the generalized
Fibonacci pseudoprimes".
5341:
4292:{\displaystyle Q^{(n+1)/2}\equiv Q\cdot Q^{(n-1)/2}\equiv Q\cdot \left({\tfrac {Q}{n}}\right){\pmod {n}}.}
2894:
3359:
Lucas pseudoprimes are: 5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, and 58519 (sequence
7623:
6830:
6684:
6611:
5766:
4443:
392:
7539:
7179:
4787:
The smallest example of a strong
Fibonacci pseudoprime is 443372888629441 = 17·31·41·43·89·97·167·331.
3510:
Lucas pseudoprimes are 989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059, and 72389 (sequence
402:
code), pages 142â152 of the book by
Crandall and Pomerance, and pages 53â74 of the book by Ribenboim.
7472:
7366:
7330:
7071:
6794:
6774:
6591:
6260:
6048:
6020:
5247:
4012:
5535:
5427:; Marjorie Bicknell (September 1974). "Some Congruences of the Fibonacci Numbers Modulo a Prime p".
5409:
5346:
1826:
7194:
7058:
7053:
7021:
6784:
6759:
6754:
6729:
6659:
6655:
6586:
6476:
6308:
6104:
6073:
2360:
1512:
7593:
1136:
7597:
7351:
7346:
7260:
7234:
7132:
7111:
6883:
6764:
6714:
6636:
6606:
6546:
6313:
6293:
6224:
5937:
5660:
5444:
5314:
5296:
5237:
5206:
5022:
4532:
3960:
3842:
3042:
502:
6481:
4436:
3409:
1906:
72:
2443:
2190:
7491:
7436:
7290:
7265:
7239:
7016:
6694:
6689:
6596:
6581:
6303:
6285:
6204:
6194:
6179:
5957:
5942:
5657:
5638:
5619:
5600:
5173:
5149:
5112:
5071:
4937:. The first pseudoprimes are then 169, 385, 741, 961, 1121, 2001, 3827, 4879, 5719, 6215 ...
4750:
4446:, say, to base 2, one can obtain very powerful probabilistic tests for primality, such as the
3004:
1396:
3907:
3874:
2409:
2327:
2294:
7527:
7320:
6906:
6878:
6868:
6860:
6744:
6709:
6704:
6671:
6365:
6328:
6219:
6214:
6209:
6199:
6171:
6058:
6010:
6005:
5962:
5901:
5641:
5580:
5558:
5436:
5351:
5306:
5255:
5196:
5096:
5012:
38:
5269:
5034:
1864:
1598:
1571:
7503:
7392:
7325:
7251:
7174:
7148:
6966:
6679:
6536:
6471:
6441:
6431:
6426:
6092:
6000:
5947:
5791:
5731:
5603:
5265:
5145:
5137:
5108:
5030:
4146:
1336:; see page 1396 of. A strong Lucas pseudoprime is also a Lucas pseudoprime (for the same (
5521:
Di Porto, Adina (1993). "Nonexistence of Even
Fibonacci Pseudoprimes of the First Kind".
3379:
2164:
5251:
2505:
7508:
7376:
7361:
7225:
7189:
7164:
7040:
7011:
6996:
6873:
6769:
6739:
6466:
6421:
6298:
5896:
5891:
5886:
5858:
5843:
5756:
5741:
5719:
5706:
5586:
5169:
5100:
5064:
5055:
3940:
3797:
3106:
3086:
3066:
2485:
2400:
2258:
1544:
1492:
1472:
1452:
1432:
1376:
1356:
1169:
379:
142:
46:
5622:
5201:
5017:
4828:. The first pseudoprimes are then 35, 169, 385, 779, 899, 961, 1121, 1189, 2419, ...
7612:
7431:
7415:
7356:
7310:
7006:
6991:
6901:
6626:
6184:
6053:
6015:
5972:
5853:
5838:
5828:
5786:
5776:
5751:
5318:
4934:
4825:
190:
17:
1861:. An extra strong Lucas pseudoprime is also a strong Lucas pseudoprime for the same
1745:{\displaystyle U_{d}\equiv 0{\pmod {n}}{\text{ and }}V_{d}\equiv \pm 2{\pmod {n}}}
7467:
7456:
7371:
7209:
7184:
7101:
7001:
6971:
6946:
6930:
6835:
6802:
6551:
6525:
6436:
6375:
5952:
5848:
5781:
5761:
5736:
265:
42:
5355:
5260:
5225:
4370:
There is a similar probability estimate for the strong Lucas probable prime test.
3538:
happens to be composite, these additional conditions may help discover that fact.
5562:
7426:
7301:
7106:
6570:
6461:
6416:
6411:
6161:
6068:
5967:
5796:
5771:
5746:
3830:, there are only five odd, composite numbers less than 10 for which congruence (
399:
7563:
7544:
6840:
6451:
5181:
5059:
4908:{\displaystyle {\text{ }}U_{n}\equiv \left({\tfrac {2}{n}}\right){\pmod {n}}}
4420:
5674:
45:
and very few composite numbers pass: in this case, criteria relative to some
7169:
7096:
7088:
6893:
6807:
5925:
5665:
5646:
5627:
5608:
5332:
F. Arnault (April 1997). "The Rabin-Monier
Theorem for Lucas Pseudoprimes".
4997:
4676:
705, 2465, 2737, 3745, 4181, 5777, 6721, 10877, 13201, 15251, ... (sequence
4506:= −1. By this definition, the Fibonacci pseudoprimes form a sequence:
2091:{\displaystyle \left({\tfrac {k}{n}}\right)\left({\tfrac {-k}{n}}\right)=-1}
3056:
sequence. As we proceed, we also compute the same, corresponding powers of
398:
Some good references are chapter 8 of the book by Bressoud and Wagon (with
378:
does not hold. These are the key facts that make Lucas sequences useful in
57:
Baillie and Wagstaff define Lucas pseudoprimes as follows: Given integers
7270:
5370:
4510:
323, 377, 1891, 3827, 4181, 5777, 6601, 6721, 8149, 10877, ... (sequence
4097:{\displaystyle Q^{(n-1)/2}\equiv \left({\tfrac {Q}{n}}\right){\pmod {n}}}
2690:{\displaystyle V_{2k}=V_{k}^{2}-2Q^{k}={\frac {V_{k}^{2}+DU_{k}^{2}}{2}}}
5310:
2276:
for squareness until after the first few search steps have all failed.)
7275:
6934:
5592:
5481:
5448:
5210:
5026:
4435:+1) is a perfect square. One can quickly detect perfect squares using
1901:
Before embarking on a probable prime test, one usually verifies that
5440:
5301:
5242:
4524:
The references of Anderson and Jacobsen below use this definition.
2291:, there are recurrence relations that enable us to quickly compute
4672:
This definition leads the Fibonacci pseudoprimes form a sequence:
4642:
This leads to an alternative definition of Fibonacci pseudoprime:
1489:) such that the number of strong Lucas pseudoprimes not exceeding
1340:) pair), but the converse is not necessarily true. Therefore, the
5070:. New York: Key College Publishing in cooperation with Springer.
3591:, then we have the following (see equation 2 on page 1392 of ):
2272:
is not square. Thus, some time can be saved by delaying testing
2268:
is square, and conversely if it does succeed, that is proof that
4396:
applications of the strong Lucas test would declare a composite
4373:
Aside from two trivial exceptions (see below), the fraction of (
3710:
of these congruences are true, then it is even more likely that
3340:
versions of the Lucas test can be implemented in a similar way.
1429:
is positive but not a square. Then there is a positive constant
7561:
7525:
7489:
7453:
7413:
7038:
6927:
6653:
6568:
6523:
6400:
6090:
6037:
5989:
5923:
5875:
5813:
5717:
5678:
3001:; this is even so it can now be divided by 2. The numerator of
2701:
Next, we can increase the subscript by 1 using the recurrences
5581:
Fibonacci Pseudoprimes, their factors, and their entry points.
5587:
Fibonacci Pseudoprimes under 2,217,967,487 and their factors.
5287:(July 2021). "Strengthening the Baillie-PSW Primality Test".
4329:) is equivalent to augmenting our Lucas test with a "base Q"
1025:
4344:
of these conditions fails to hold, then we have proved that
4837:
4699:
4678:
4512:
4336:
Additional congruence conditions that must be satisfied if
3512:
3361:
3325:
744:
3937:, and we can easily save the previously computed power of
250:{\displaystyle \delta (n)=n-\left({\tfrac {D}{n}}\right).}
1126:{\displaystyle \delta (n)=n-\left({\tfrac {D}{n}}\right)}
4400:
to be probably prime with a probability at most (4/15).
2880:{\displaystyle V_{2k+1}=(D\cdot U_{2k}+P\cdot V_{2k})/2}
4367:
to be probably prime with a probability at most (1/4).
1648:
is a strong Lucas pseudoprime for a set of parameters (
1637:, for example, the least strong Lucas pseudoprime with
1344:
test is a more stringent primality test than equation (
4874:
4255:
4120:
4063:
3763:
3557:
3460:
3048:.) Observe that, for each term that we compute in the
2118:
2059:
2039:
1993:
1924:
1108:
922:
885:{\displaystyle U_{20}=6616217487\equiv 0{\pmod {19}}.}
795:
673:
is prime or it is a Lucas pseudoprime. If congruence (
524:
476:
229:
164:
5502:
P. S. Bruckman (1994). "Lucas Pseudoprimes are odd".
4850:
4565:
4171:
4114:
4024:
3963:
3943:
3910:
3877:
3845:
3757:
3602:
3551:
3454:
3412:
3382:
3273:
By the end of the calculation, we will have computed
3109:
3089:
3069:
3007:
2949:
2897:
2799:
2710:
2591:
2534:
2508:
2488:
2446:
2412:
2363:
2330:
2297:
2193:
2167:
2112:
2033:
1987:
1918:
1867:
1829:
1764:
1669:
1601:
1574:
1547:
1515:
1495:
1475:
1455:
1435:
1399:
1379:
1359:
1269:
1214:
1172:
1139:
1081:
1013:) pair. In fact, 119 is the smallest pseudoprime for
956:
916:
840:
789:
578:
518:
470:
283:
202:
158:
75:
5462:
David Singmaster (1983). "Some Lucas Pseudoprimes".
1005:
However, 119 = 7·17 is not prime, so 119 is a Lucas
7385:
7339:
7299:
7250:
7224:
7157:
7141:
7120:
7087:
7052:
6892:
6859:
6816:
6793:
6670:
6358:
6349:
6327:
6284:
6246:
6237:
6170:
6112:
6103:
5091:
5089:
5087:
3052:sequence, we compute the corresponding term in the
1912:We choose a Lucas sequence where the Jacobi symbol
1813:{\displaystyle V_{d\cdot 2^{r}}\equiv 0{\pmod {n}}}
1318:{\displaystyle V_{d\cdot 2^{r}}\equiv 0{\pmod {n}}}
5063:
4907:
4742:is a strong Fibonacci pseudoprime if and only if:
4618:
4291:
4153:is prime, this is the same as the Jacobi symbol).
4137:
4096:
3996:
3949:
3929:
3896:
3860:
3780:
3649:
3583:
3486:
3440:
3394:
3115:
3095:
3075:
3029:
2993:
2935:
2879:
2784:
2689:
2576:
2517:
2494:
2471:
2431:
2391:
2349:
2316:
2225:
2179:
2141:
2090:
2019:
1950:
1885:
1853:
1812:
1744:
1614:
1587:
1553:
1533:
1501:
1481:
1461:
1441:
1421:
1385:
1365:
1317:
1249:
1178:
1158:
1125:
994:
939:
895:Therefore, 19 is a Lucas probable prime for this (
884:
812:
622:
553:
493:
331:{\displaystyle U_{\delta (n)}\equiv 0{\pmod {n}}.}
330:
249:
181:
103:
27:Probabilistic test for the primality of an integer
3714:is prime than if we had checked only congruence (
3234:. We also compute the same-numbered terms in the
2785:{\displaystyle U_{2k+1}=(P\cdot U_{2k}+V_{2k})/2}
272:, then the following congruence condition holds:
1630:-Lucas sequence, the pseudoprimes can be called
1393:be relatively prime positive integers for which
669:is a Lucas probable prime. In this case, either
554:{\displaystyle \left({\tfrac {D}{n}}\right)=-1,}
4352:Comparison with the MillerâRabin primality test
3800:). If we use this enhanced method for choosing
3584:{\displaystyle \left({\tfrac {D}{n}}\right)=-1}
3487:{\displaystyle \left({\tfrac {D}{n}}\right)=-1}
2020:{\displaystyle \left({\tfrac {D}{n}}\right)=-1}
1951:{\displaystyle \left({\tfrac {D}{n}}\right)=-1}
940:{\displaystyle \left({\tfrac {13}{119}}\right)}
391:) represents one of two congruences defining a
4619:{\displaystyle V_{n}(P,Q)\equiv P{\pmod {n}}.}
2142:{\displaystyle \left({\tfrac {D}{n}}\right)=0}
813:{\displaystyle \left({\tfrac {13}{19}}\right)}
460:A Lucas probable prime test is most useful if
5690:
4481:) sequence represents the Fibonacci numbers.
4442:By combining a Lucas pseudoprime test with a
3812:composite less than 10 for which congruence (
1981:in the sequence 5, â7, 9, â11, ... such that
995:{\displaystyle U_{120}\equiv 0{\pmod {119}}.}
8:
4738:. It follows that an odd composite integer
4138:{\displaystyle \left({\tfrac {Q}{n}}\right)}
3781:{\displaystyle \left({\tfrac {D}{n}}\right)}
3723:If Selfridge's method (above) for choosing
3650:{\displaystyle V_{n+1}\equiv 2Q{\pmod {n}}.}
3355:are chosen as described above, the first 10
3130:We use the bits of the binary expansion of
1977:is to use trial and error to find the first
494:{\displaystyle \left({\tfrac {D}{n}}\right)}
182:{\displaystyle \left({\tfrac {D}{n}}\right)}
5132:
5130:
5128:
4940:A third definition uses equation (5) with (
4340:is prime are described in Section 6 of. If
2525:in one step using the recurrence relations
899:) pair. In this case 19 is prime, so it is
715:is composite, we gain more confidence that
711:, then unless one of the tests proves that
623:{\displaystyle U_{n+1}\equiv 0{\pmod {n}}.}
41:integers that pass certain tests which all
7558:
7522:
7486:
7450:
7410:
7084:
7049:
7035:
6924:
6667:
6650:
6565:
6520:
6397:
6355:
6243:
6109:
6100:
6087:
6034:
5991:Possessing a specific set of other numbers
5986:
5920:
5872:
5810:
5714:
5697:
5683:
5675:
5105:Prime numbers: A computational perspective
5050:
5048:
5046:
5044:
3702:) is false, this constitutes a proof that
652:) is false, this constitutes a proof that
5557:. Vol. 5. Kluwer. pp. 459â464.
5548:
5546:
5534:
5408:
5345:
5300:
5259:
5241:
5200:
5016:
4889:
4873:
4860:
4851:
4849:
4597:
4570:
4564:
4492:not divisible by 5 for which congruence (
4427:is easy to factor, because in this case,
4403:There are two trivial exceptions. One is
4270:
4254:
4231:
4215:
4192:
4176:
4170:
4119:
4113:
4078:
4062:
4045:
4029:
4023:
3984:
3968:
3962:
3942:
3915:
3909:
3882:
3876:
3844:
3762:
3756:
3680:was computed in the process of computing
3628:
3607:
3601:
3556:
3550:
3524:Checking additional congruence conditions
3459:
3453:
3423:
3411:
3381:
3108:
3088:
3068:
3012:
3006:
2976:
2960:
2948:
2924:
2908:
2896:
2869:
2857:
2835:
2804:
2798:
2774:
2762:
2746:
2715:
2709:
2675:
2670:
2654:
2649:
2642:
2633:
2617:
2612:
2596:
2590:
2568:
2555:
2539:
2533:
2507:
2487:
2451:
2445:
2417:
2411:
2374:
2362:
2335:
2329:
2302:
2296:
2215:
2192:
2166:
2117:
2111:
2058:
2038:
2032:
1992:
1986:
1923:
1917:
1866:
1828:
1794:
1780:
1769:
1763:
1726:
1711:
1702:
1686:
1674:
1668:
1606:
1600:
1579:
1573:
1546:
1514:
1494:
1474:
1454:
1434:
1404:
1398:
1378:
1358:
1299:
1285:
1274:
1268:
1231:
1219:
1213:
1171:
1150:
1138:
1107:
1080:
973:
961:
955:
921:
915:
863:
845:
839:
794:
788:
601:
583:
577:
523:
517:
475:
469:
309:
288:
282:
228:
201:
163:
157:
86:
74:
5593:Pseudoprime Statistics, Tables, and Data
5369:Adina Di Porto; Piero Filipponi (1989).
4389:to be probably prime is at most (4/15).
2482:First, we can double the subscript from
1897:Implementing a Lucas probable prime test
1250:{\displaystyle U_{d}\equiv 0{\pmod {n}}}
1205:) = 1, satisfying one of the conditions
1057:= â1, the smallest Lucas pseudoprime to
831:= 6616217487 = 19·348221973 and we have
5066:A Course in Computational Number Theory
4987:
4985:
4983:
4981:
4957:
4707:) under the first definition given by (
4488:is often defined as a composite number
2577:{\displaystyle U_{2k}=U_{k}\cdot V_{k}}
1641:= 1, 2, 3, ... are 4181, 169, 119, ...
4979:
4977:
4975:
4973:
4971:
4969:
4967:
4965:
4963:
4961:
3904:is computed during the calculation of
2994:{\displaystyle P\cdot U_{2k}+V_{2k}+n}
1660:= 1, satisfying one of the conditions
464:is chosen such that the Jacobi symbol
406:Lucas probable primes and pseudoprimes
4799:may be defined as a composite number
3142:sequence to compute. For example, if
2401:Lucas sequence § Other relations
687:to be prime (this justifies the term
7:
5371:"More on the Fibonacci Pseudoprimes"
5142:The New Book of Prime Number Records
4831:This differs from the definition in
4556:
4162:
3593:
3528:If we have checked that congruence (
3406:= 3, 4, 5, 6, ..., until a value of
3037:is handled in the same way. (Adding
2936:{\displaystyle P\cdot U_{2k}+V_{2k}}
2237:has no prime factors in common with
2106:have a prime factor in common, then
569:
274:
5482:"Pseudoprime Statistics and Tables"
4897:
4605:
4323:) = 1 then testing for congruence (
4278:
4086:
3636:
3448:is found so that the Jacobi symbol
2264:(This search will never succeed if
1802:
1734:
1694:
1307:
1239:
981:
871:
609:
317:
53:Baillie-Wagstaff-Lucas pseudoprimes
3798:Lucas sequence-Algebraic relations
2233:. It is a good idea to check that
1197:) pair is an odd composite number
1028:that, in order to check equation (
432:) above is true (see, page 1398).
25:
5555:Applications of Fibonacci Numbers
5202:10.1090/S0025-5718-1980-0572872-7
5018:10.1090/S0025-5718-1980-0583518-6
3494:. With this method for selecting
1632:strong Lucas pseudoprime in base
1042:need to compute all of the first
7592:
7200:Perfect digit-to-digit invariant
5661:"Extra Strong Lucas Pseudoprime"
4890:
4598:
4331:SolovayâStrassen primality test
4271:
4079:
3739:= −1, then we can adjust
3629:
1795:
1727:
1687:
1300:
1232:
974:
947:= −1, and we can compute
864:
602:
310:
5283:Robert Baillie; Andrew Fiori;
4901:
4891:
4690:which are also referred to as
4609:
4599:
4588:
4576:
4319:cannot be prime. Provided GCD(
4282:
4272:
4228:
4216:
4189:
4177:
4090:
4080:
4042:
4030:
3981:
3969:
3640:
3630:
2866:
2822:
2771:
2733:
2386:
2367:
2212:
2200:
2153:values, the average number of
1880:
1868:
1854:{\displaystyle 0\leq r<s-1}
1806:
1796:
1738:
1728:
1698:
1688:
1646:extra strong Lucas pseudoprime
1311:
1301:
1243:
1233:
1091:
1085:
985:
975:
875:
865:
613:
603:
321:
311:
298:
292:
212:
206:
152:be a positive integer and let
1:
6039:Expressible via specific sums
5356:10.1090/s0025-5718-97-00836-3
5261:10.1090/S0025-5718-00-01197-2
3295:). We then check congruence (
2392:{\displaystyle O(\log _{2}n)}
1973:, one technique for choosing
1534:{\displaystyle c\cdot \log x}
5563:10.1007/978-94-011-2058-6_45
4718:strong Fibonacci pseudoprime
3178:= 44. Therefore, we compute
1159:{\displaystyle d\cdot 2^{s}}
457:) is true (see, page 1391).
7128:Multiplicative digital root
5182:"The pseudoprimes to 25·10"
4926:) = (2, â1) again defining
4805:
4726:
4709:
4656:
4494:
4385:) that declare a composite
4361:MillerâRabin primality test
4325:
3997:{\displaystyle Q^{(n+1)/2}}
3861:{\displaystyle Q\neq \pm 1}
3832:
3814:
3716:
3698:
3692:
3530:
3301:) using our known value of
3297:
3041:does not change the result
1346:
1030:
675:
661:
648:
563:
453:
428:
387:
7640:
5642:"Strong Lucas Pseudoprime"
5334:Mathematics of Computation
5289:Mathematics of Computation
5230:Mathematics of Computation
5189:Mathematics of Computation
5005:Mathematics of Computation
4448:BaillieâPSW primality test
4419:+2) is the product of two
3808:, then 913 = 11·83 is the
3441:{\displaystyle D=P^{2}-4Q}
2245:. This method of choosing
906:For the next example, let
691:prime), but this does not
507:BaillieâPSW primality test
104:{\displaystyle D=P^{2}-4Q}
7588:
7571:
7557:
7535:
7521:
7499:
7485:
7463:
7449:
7422:
7409:
7205:Perfect digital invariant
7048:
7034:
6942:
6923:
6780:Superior highly composite
6666:
6649:
6577:
6564:
6532:
6519:
6407:
6396:
6099:
6086:
6044:
6033:
5996:
5985:
5933:
5919:
5882:
5871:
5824:
5809:
5727:
5713:
5464:Abstracts Amer. Math. Soc
4841:which may be written as:
4817:= −1; the sequence
4554:) = 1, then we also have
3063:At each stage, we reduce
2472:{\displaystyle V_{1}=P=1}
2226:{\displaystyle Q=(1-D)/4}
2149:). With this sequence of
1071:Strong Lucas pseudoprimes
6818:Euler's totient function
6602:EulerâJacobi pseudoprime
5877:Other polynomial numbers
5226:"Frobenius Pseudoprimes"
4160:is prime, we must have,
3376:Lucas pseudoprimes, set
3030:{\displaystyle V_{2k+1}}
2943:is odd, replace it with
1626:-Fibonacci sequence and
1422:{\displaystyle P^{2}-4Q}
1191:strong Lucas pseudoprime
783:= 19. The Jacobi symbol
354:If this congruence does
6632:SomerâLucas pseudoprime
6622:LucasâCarmichael number
6457:Lazy caterer's sequence
5604:"Fibonacci Pseudoprime"
5285:Samuel S. Wagstaff, Jr.
5178:Samuel S. Wagstaff, Jr.
4994:Samuel S. Wagstaff, Jr.
4535:base 2. However, when
4407:= 9. The other is when
3930:{\displaystyle U_{n+1}}
3897:{\displaystyle Q^{n+1}}
3372:To calculate a list of
2432:{\displaystyle U_{1}=1}
2350:{\displaystyle V_{n+1}}
2317:{\displaystyle U_{n+1}}
374:, then this congruence
141:) be the corresponding
6507:WedderburnâEtherington
5907:Lucky numbers of Euler
4909:
4724:for which congruence (
4720:is a composite number
4654:for which congruence (
4650:is a composite number
4620:
4454:Fibonacci pseudoprimes
4293:
4139:
4098:
3998:
3951:
3931:
3898:
3862:
3782:
3690:If either congruence (
3651:
3585:
3488:
3442:
3396:
3117:
3097:
3077:
3031:
2995:
2937:
2881:
2786:
2691:
2578:
2519:
2496:
2473:
2433:
2393:
2351:
2318:
2227:
2181:
2143:
2092:
2021:
1952:
1887:
1855:
1814:
1746:
1616:
1589:
1555:
1535:
1503:
1483:
1463:
1443:
1423:
1387:
1367:
1319:
1251:
1180:
1160:
1127:
996:
941:
886:
814:
734:= 13, the sequence of
624:
555:
495:
332:
251:
183:
105:
35:Fibonacci pseudoprimes
6795:Prime omega functions
6612:Frobenius pseudoprime
6402:Combinatorial numbers
6271:Centered dodecahedral
6064:Primary pseudoperfect
5224:Jon Grantham (2001).
4910:
4809:) above is true with
4648:Fibonacci pseudoprime
4621:
4486:Fibonacci pseudoprime
4444:Fermat primality test
4294:
4140:
4099:
3999:
3952:
3932:
3899:
3863:
3788:remain unchanged and
3783:
3652:
3586:
3489:
3443:
3397:
3238:sequence, along with
3118:
3098:
3078:
3032:
2996:
2938:
2882:
2787:
2692:
2579:
2520:
2497:
2474:
2434:
2394:
2352:
2319:
2228:
2182:
2144:
2093:
2022:
1953:
1888:
1886:{\displaystyle (P,Q)}
1856:
1815:
1747:
1617:
1615:{\displaystyle V_{n}}
1590:
1588:{\displaystyle U_{n}}
1556:
1536:
1504:
1484:
1464:
1444:
1424:
1388:
1368:
1320:
1252:
1181:
1161:
1128:
997:
942:
903:a Lucas pseudoprime.
887:
815:
625:
556:
496:
443:) pair is a positive
393:Frobenius pseudoprime
333:
268:that does not divide
252:
184:
106:
18:Fibonacci pseudoprime
7254:-composition related
7054:Arithmetic functions
6656:Arithmetic functions
6592:Elliptic pseudoprime
6276:Centered icosahedral
6256:Centered tetrahedral
5429:Mathematics Magazine
4998:"Lucas Pseudoprimes"
4848:
4803:for which equation (
4563:
4363:declare a composite
4359:applications of the
4169:
4112:
4022:
3961:
3941:
3908:
3875:
3843:
3755:
3600:
3549:
3545:is an odd prime and
3452:
3410:
3380:
3107:
3087:
3067:
3005:
2947:
2895:
2797:
2708:
2589:
2532:
2506:
2486:
2444:
2410:
2361:
2328:
2295:
2191:
2165:
2110:
2031:
1985:
1916:
1865:
1827:
1762:
1667:
1599:
1572:
1561:sufficiently large.
1545:
1513:
1493:
1473:
1453:
1433:
1397:
1377:
1357:
1267:
1212:
1170:
1137:
1079:
954:
914:
838:
787:
576:
516:
468:
451:for which equation (
426:for which equation (
412:Lucas probable prime
281:
200:
156:
73:
7180:Kaprekar's constant
6700:Colossally abundant
6587:Catalan pseudoprime
6487:SchröderâHipparchus
6266:Centered octahedral
6142:Centered heptagonal
6132:Centered pentagonal
6122:Centered triangular
5722:and related numbers
5623:"Lucas Pseudoprime"
5585:Anderson, Peter G.
5579:Anderson, Peter G.
5523:Fibonacci Quarterly
5504:Fibonacci Quarterly
5397:Fibonacci Quarterly
5378:Fibonacci Quarterly
5252:2001MaCom..70..873G
5097:Richard E. Crandall
4734:= −1 and all
4011:is prime, then, by
3395:{\displaystyle Q=1}
3103:, and the power of
2680:
2659:
2622:
2180:{\displaystyle P=1}
1328:for some 0 ≤
1061:= 1, 2, 3, ... are
820:is −1, so ÎŽ(
7598:Mathematics portal
7540:Aronson's sequence
7286:SmarandacheâWellin
7043:-dependent numbers
6750:Primitive abundant
6637:Strong pseudoprime
6627:Perrin pseudoprime
6607:Fermat pseudoprime
6547:Wolstenholme prime
6371:Squared triangular
6157:Centered decagonal
6152:Centered nonagonal
6147:Centered octagonal
6137:Centered hexagonal
5658:Weisstein, Eric W.
5639:Weisstein, Eric W.
5620:Weisstein, Eric W.
5601:Weisstein, Eric W.
5470:(83Tâ10â146): 197.
5425:V. E. Hoggatt, Jr.
5295:(330): 1931â1955.
5195:(151): 1003â1026.
5011:(152): 1391â1417.
4905:
4883:
4775:) for every prime
4616:
4533:Fermat pseudoprime
4439:for square roots.
4289:
4264:
4135:
4129:
4094:
4072:
3994:
3947:
3927:
3894:
3858:
3778:
3772:
3647:
3581:
3566:
3484:
3469:
3438:
3392:
3113:
3093:
3073:
3027:
2991:
2933:
2877:
2782:
2687:
2666:
2645:
2608:
2574:
2518:{\displaystyle 2k}
2515:
2492:
2469:
2429:
2389:
2347:
2314:
2223:
2177:
2139:
2127:
2088:
2073:
2048:
2017:
2002:
1948:
1933:
1909:for square roots.
1883:
1851:
1810:
1742:
1612:
1585:
1551:
1531:
1499:
1479:
1459:
1439:
1419:
1383:
1363:
1315:
1247:
1176:
1156:
1123:
1117:
992:
937:
931:
882:
810:
804:
620:
551:
533:
505:test, such as the
503:strong pseudoprime
491:
485:
328:
247:
238:
179:
173:
101:
31:Lucas pseudoprimes
7619:Fibonacci numbers
7606:
7605:
7584:
7583:
7553:
7552:
7517:
7516:
7481:
7480:
7445:
7444:
7405:
7404:
7401:
7400:
7220:
7219:
7030:
7029:
6919:
6918:
6915:
6914:
6861:Aliquot sequences
6672:Divisor functions
6645:
6644:
6617:Lucas pseudoprime
6597:Euler pseudoprime
6582:Carmichael number
6560:
6559:
6515:
6514:
6392:
6391:
6388:
6387:
6384:
6383:
6345:
6344:
6233:
6232:
6190:Square triangular
6082:
6081:
6029:
6028:
5981:
5980:
5915:
5914:
5867:
5866:
5805:
5804:
5311:10.1090/mcom/3616
5174:John L. Selfridge
5077:978-1-930190-10-8
4882:
4854:
4791:Pell pseudoprimes
4751:Carmichael number
4640:
4639:
4546:is prime and GCD(
4313:
4312:
4263:
4128:
4071:
4013:Euler's criterion
3950:{\displaystyle Q}
3771:
3706:is not prime. If
3671:
3670:
3565:
3468:
3116:{\displaystyle Q}
3096:{\displaystyle V}
3076:{\displaystyle U}
2685:
2495:{\displaystyle k}
2403:. To start off,
2257:was suggested by
2126:
2072:
2047:
2001:
1932:
1705:
1554:{\displaystyle x}
1502:{\displaystyle x}
1482:{\displaystyle Q}
1462:{\displaystyle P}
1442:{\displaystyle c}
1386:{\displaystyle Q}
1366:{\displaystyle P}
1179:{\displaystyle d}
1116:
1046:+ 1 terms in the
930:
803:
644:
643:
532:
484:
437:Lucas pseudoprime
422:positive integer
380:primality testing
352:
351:
237:
172:
16:(Redirected from
7631:
7596:
7559:
7528:Natural language
7523:
7487:
7455:Generated via a
7451:
7411:
7316:Digit-reassembly
7281:Self-descriptive
7085:
7050:
7036:
6987:LucasâCarmichael
6977:Harmonic divisor
6925:
6851:Sparsely totient
6826:Highly cototient
6735:Multiply perfect
6725:Highly composite
6668:
6651:
6566:
6521:
6502:Telephone number
6398:
6356:
6337:Square pyramidal
6319:Stella octangula
6244:
6110:
6101:
6093:Figurate numbers
6088:
6035:
5987:
5921:
5873:
5811:
5715:
5699:
5692:
5685:
5676:
5671:
5670:
5652:
5651:
5633:
5632:
5614:
5613:
5567:
5566:
5550:
5541:
5540:
5538:
5518:
5512:
5511:
5499:
5493:
5492:
5490:
5488:
5478:
5472:
5471:
5459:
5453:
5452:
5421:
5415:
5414:
5412:
5392:
5386:
5385:
5375:
5366:
5360:
5359:
5349:
5340:(218): 869â881.
5329:
5323:
5322:
5304:
5280:
5274:
5273:
5263:
5245:
5236:(234): 873â891.
5221:
5215:
5214:
5204:
5186:
5166:
5160:
5159:
5134:
5123:
5122:
5107:(2nd ed.).
5093:
5082:
5081:
5069:
5052:
5039:
5038:
5020:
5002:
4996:(October 1980).
4992:Robert Baillie;
4989:
4914:
4912:
4911:
4906:
4904:
4888:
4884:
4875:
4865:
4864:
4855:
4852:
4840:
4797:Pell pseudoprime
4763:− 1) or 2(
4702:
4681:
4634:
4625:
4623:
4622:
4617:
4612:
4575:
4574:
4557:
4515:
4466:= −1, the
4381:) pairs (modulo
4307:
4298:
4296:
4295:
4290:
4285:
4269:
4265:
4256:
4240:
4239:
4235:
4201:
4200:
4196:
4163:
4144:
4142:
4141:
4136:
4134:
4130:
4121:
4103:
4101:
4100:
4095:
4093:
4077:
4073:
4064:
4054:
4053:
4049:
4003:
4001:
4000:
3995:
3993:
3992:
3988:
3956:
3954:
3953:
3948:
3936:
3934:
3933:
3928:
3926:
3925:
3903:
3901:
3900:
3895:
3893:
3892:
3867:
3865:
3864:
3859:
3787:
3785:
3784:
3779:
3777:
3773:
3764:
3735:happened to set
3665:
3656:
3654:
3653:
3648:
3643:
3618:
3617:
3594:
3590:
3588:
3587:
3582:
3571:
3567:
3558:
3515:
3493:
3491:
3490:
3485:
3474:
3470:
3461:
3447:
3445:
3444:
3439:
3428:
3427:
3401:
3399:
3398:
3393:
3364:
3328:
3122:
3120:
3119:
3114:
3102:
3100:
3099:
3094:
3082:
3080:
3079:
3074:
3036:
3034:
3033:
3028:
3026:
3025:
3000:
2998:
2997:
2992:
2984:
2983:
2968:
2967:
2942:
2940:
2939:
2934:
2932:
2931:
2916:
2915:
2886:
2884:
2883:
2878:
2873:
2865:
2864:
2843:
2842:
2818:
2817:
2791:
2789:
2788:
2783:
2778:
2770:
2769:
2754:
2753:
2729:
2728:
2696:
2694:
2693:
2688:
2686:
2681:
2679:
2674:
2658:
2653:
2643:
2638:
2637:
2621:
2616:
2604:
2603:
2583:
2581:
2580:
2575:
2573:
2572:
2560:
2559:
2547:
2546:
2524:
2522:
2521:
2516:
2501:
2499:
2498:
2493:
2478:
2476:
2475:
2470:
2456:
2455:
2438:
2436:
2435:
2430:
2422:
2421:
2398:
2396:
2395:
2390:
2379:
2378:
2356:
2354:
2353:
2348:
2346:
2345:
2323:
2321:
2320:
2315:
2313:
2312:
2232:
2230:
2229:
2224:
2219:
2186:
2184:
2183:
2178:
2148:
2146:
2145:
2140:
2132:
2128:
2119:
2097:
2095:
2094:
2089:
2078:
2074:
2068:
2060:
2053:
2049:
2040:
2026:
2024:
2023:
2018:
2007:
2003:
1994:
1957:
1955:
1954:
1949:
1938:
1934:
1925:
1892:
1890:
1889:
1884:
1860:
1858:
1857:
1852:
1819:
1817:
1816:
1811:
1809:
1787:
1786:
1785:
1784:
1751:
1749:
1748:
1743:
1741:
1716:
1715:
1706:
1703:
1701:
1679:
1678:
1621:
1619:
1618:
1613:
1611:
1610:
1594:
1592:
1591:
1586:
1584:
1583:
1560:
1558:
1557:
1552:
1540:
1538:
1537:
1532:
1509:is greater than
1508:
1506:
1505:
1500:
1488:
1486:
1485:
1480:
1468:
1466:
1465:
1460:
1448:
1446:
1445:
1440:
1428:
1426:
1425:
1420:
1409:
1408:
1392:
1390:
1389:
1384:
1372:
1370:
1369:
1364:
1324:
1322:
1321:
1316:
1314:
1292:
1291:
1290:
1289:
1256:
1254:
1253:
1248:
1246:
1224:
1223:
1185:
1183:
1182:
1177:
1165:
1163:
1162:
1157:
1155:
1154:
1132:
1130:
1129:
1124:
1122:
1118:
1109:
1001:
999:
998:
993:
988:
966:
965:
946:
944:
943:
938:
936:
932:
923:
891:
889:
888:
883:
878:
850:
849:
819:
817:
816:
811:
809:
805:
796:
747:
730:= −1, and
679:) is true, then
665:) is true, then
638:
629:
627:
626:
621:
616:
594:
593:
570:
560:
558:
557:
552:
538:
534:
525:
500:
498:
497:
492:
490:
486:
477:
385:The congruence (
346:
337:
335:
334:
329:
324:
302:
301:
275:
256:
254:
253:
248:
243:
239:
230:
188:
186:
185:
180:
178:
174:
165:
110:
108:
107:
102:
91:
90:
21:
7639:
7638:
7634:
7633:
7632:
7630:
7629:
7628:
7609:
7608:
7607:
7602:
7580:
7576:Strobogrammatic
7567:
7549:
7531:
7513:
7495:
7477:
7459:
7441:
7418:
7397:
7381:
7340:Divisor-related
7335:
7295:
7246:
7216:
7153:
7137:
7116:
7083:
7056:
7044:
7026:
6938:
6937:related numbers
6911:
6888:
6855:
6846:Perfect totient
6812:
6789:
6720:Highly abundant
6662:
6641:
6573:
6556:
6528:
6511:
6497:Stirling second
6403:
6380:
6341:
6323:
6280:
6229:
6166:
6127:Centered square
6095:
6078:
6040:
6025:
5992:
5977:
5929:
5928:defined numbers
5911:
5878:
5863:
5834:Double Mersenne
5820:
5801:
5723:
5709:
5707:natural numbers
5703:
5656:
5655:
5637:
5636:
5618:
5617:
5599:
5598:
5591:Jacobsen, Dana
5576:
5571:
5570:
5552:
5551:
5544:
5536:10.1.1.376.2601
5520:
5519:
5515:
5501:
5500:
5496:
5486:
5484:
5480:
5479:
5475:
5461:
5460:
5456:
5441:10.2307/2689212
5423:
5422:
5418:
5410:10.1.1.388.4993
5394:
5393:
5389:
5373:
5368:
5367:
5363:
5347:10.1.1.192.4789
5331:
5330:
5326:
5282:
5281:
5277:
5223:
5222:
5218:
5184:
5168:
5167:
5163:
5156:
5146:Springer-Verlag
5138:Paulo Ribenboim
5136:
5135:
5126:
5119:
5109:Springer-Verlag
5095:
5094:
5085:
5078:
5054:
5053:
5042:
5000:
4991:
4990:
4959:
4954:
4931:
4869:
4856:
4846:
4845:
4832:
4824:then being the
4822:
4793:
4698:
4677:
4632:
4566:
4561:
4560:
4511:
4471:
4456:
4437:Newton's method
4354:
4305:
4250:
4211:
4172:
4167:
4166:
4147:Legendre symbol
4115:
4110:
4109:
4058:
4025:
4020:
4019:
3964:
3959:
3958:
3939:
3938:
3911:
3906:
3905:
3878:
3873:
3872:
3841:
3840:
3758:
3753:
3752:
3685:
3678:
3663:
3603:
3598:
3597:
3552:
3547:
3546:
3526:
3511:
3506:, the first 10
3455:
3450:
3449:
3419:
3408:
3407:
3378:
3377:
3360:
3324:
3306:
3285:
3278:
3233:
3226:
3219:
3212:
3205:
3198:
3191:
3184:
3177:
3173:
3169:
3165:
3161:
3157:
3153:
3149:
3105:
3104:
3085:
3084:
3065:
3064:
3008:
3003:
3002:
2972:
2956:
2945:
2944:
2920:
2904:
2893:
2892:
2853:
2831:
2800:
2795:
2794:
2758:
2742:
2711:
2706:
2705:
2644:
2629:
2592:
2587:
2586:
2564:
2551:
2535:
2530:
2529:
2504:
2503:
2484:
2483:
2447:
2442:
2441:
2413:
2408:
2407:
2370:
2359:
2358:
2331:
2326:
2325:
2298:
2293:
2292:
2189:
2188:
2163:
2162:
2113:
2108:
2107:
2061:
2054:
2034:
2029:
2028:
1988:
1983:
1982:
1919:
1914:
1913:
1907:Newton's method
1899:
1863:
1862:
1825:
1824:
1776:
1765:
1760:
1759:
1707:
1704: and
1670:
1665:
1664:
1602:
1597:
1596:
1575:
1570:
1569:
1543:
1542:
1511:
1510:
1491:
1490:
1471:
1470:
1451:
1450:
1431:
1430:
1400:
1395:
1394:
1375:
1374:
1355:
1354:
1281:
1270:
1265:
1264:
1215:
1210:
1209:
1168:
1167:
1146:
1135:
1134:
1103:
1077:
1076:
1073:
957:
952:
951:
917:
912:
911:
910:= 119. We have
841:
836:
835:
829:
790:
785:
784:
774:
767:
760:
753:
739:
659:If congruence (
646:If congruence (
636:
579:
574:
573:
561:then equation (
519:
514:
513:
471:
466:
465:
408:
344:
284:
279:
278:
224:
198:
197:
159:
154:
153:
143:Lucas sequences
131:
116:
82:
71:
70:
55:
28:
23:
22:
15:
12:
11:
5:
7637:
7635:
7627:
7626:
7621:
7611:
7610:
7604:
7603:
7601:
7600:
7589:
7586:
7585:
7582:
7581:
7579:
7578:
7572:
7569:
7568:
7562:
7555:
7554:
7551:
7550:
7548:
7547:
7542:
7536:
7533:
7532:
7526:
7519:
7518:
7515:
7514:
7512:
7511:
7509:Sorting number
7506:
7504:Pancake number
7500:
7497:
7496:
7490:
7483:
7482:
7479:
7478:
7476:
7475:
7470:
7464:
7461:
7460:
7454:
7447:
7446:
7443:
7442:
7440:
7439:
7434:
7429:
7423:
7420:
7419:
7416:Binary numbers
7414:
7407:
7406:
7403:
7402:
7399:
7398:
7396:
7395:
7389:
7387:
7383:
7382:
7380:
7379:
7374:
7369:
7364:
7359:
7354:
7349:
7343:
7341:
7337:
7336:
7334:
7333:
7328:
7323:
7318:
7313:
7307:
7305:
7297:
7296:
7294:
7293:
7288:
7283:
7278:
7273:
7268:
7263:
7257:
7255:
7248:
7247:
7245:
7244:
7243:
7242:
7231:
7229:
7226:P-adic numbers
7222:
7221:
7218:
7217:
7215:
7214:
7213:
7212:
7202:
7197:
7192:
7187:
7182:
7177:
7172:
7167:
7161:
7159:
7155:
7154:
7152:
7151:
7145:
7143:
7142:Coding-related
7139:
7138:
7136:
7135:
7130:
7124:
7122:
7118:
7117:
7115:
7114:
7109:
7104:
7099:
7093:
7091:
7082:
7081:
7080:
7079:
7077:Multiplicative
7074:
7063:
7061:
7046:
7045:
7041:Numeral system
7039:
7032:
7031:
7028:
7027:
7025:
7024:
7019:
7014:
7009:
7004:
6999:
6994:
6989:
6984:
6979:
6974:
6969:
6964:
6959:
6954:
6949:
6943:
6940:
6939:
6928:
6921:
6920:
6917:
6916:
6913:
6912:
6910:
6909:
6904:
6898:
6896:
6890:
6889:
6887:
6886:
6881:
6876:
6871:
6865:
6863:
6857:
6856:
6854:
6853:
6848:
6843:
6838:
6833:
6831:Highly totient
6828:
6822:
6820:
6814:
6813:
6811:
6810:
6805:
6799:
6797:
6791:
6790:
6788:
6787:
6782:
6777:
6772:
6767:
6762:
6757:
6752:
6747:
6742:
6737:
6732:
6727:
6722:
6717:
6712:
6707:
6702:
6697:
6692:
6687:
6685:Almost perfect
6682:
6676:
6674:
6664:
6663:
6654:
6647:
6646:
6643:
6642:
6640:
6639:
6634:
6629:
6624:
6619:
6614:
6609:
6604:
6599:
6594:
6589:
6584:
6578:
6575:
6574:
6569:
6562:
6561:
6558:
6557:
6555:
6554:
6549:
6544:
6539:
6533:
6530:
6529:
6524:
6517:
6516:
6513:
6512:
6510:
6509:
6504:
6499:
6494:
6492:Stirling first
6489:
6484:
6479:
6474:
6469:
6464:
6459:
6454:
6449:
6444:
6439:
6434:
6429:
6424:
6419:
6414:
6408:
6405:
6404:
6401:
6394:
6393:
6390:
6389:
6386:
6385:
6382:
6381:
6379:
6378:
6373:
6368:
6362:
6360:
6353:
6347:
6346:
6343:
6342:
6340:
6339:
6333:
6331:
6325:
6324:
6322:
6321:
6316:
6311:
6306:
6301:
6296:
6290:
6288:
6282:
6281:
6279:
6278:
6273:
6268:
6263:
6258:
6252:
6250:
6241:
6235:
6234:
6231:
6230:
6228:
6227:
6222:
6217:
6212:
6207:
6202:
6197:
6192:
6187:
6182:
6176:
6174:
6168:
6167:
6165:
6164:
6159:
6154:
6149:
6144:
6139:
6134:
6129:
6124:
6118:
6116:
6107:
6097:
6096:
6091:
6084:
6083:
6080:
6079:
6077:
6076:
6071:
6066:
6061:
6056:
6051:
6045:
6042:
6041:
6038:
6031:
6030:
6027:
6026:
6024:
6023:
6018:
6013:
6008:
6003:
5997:
5994:
5993:
5990:
5983:
5982:
5979:
5978:
5976:
5975:
5970:
5965:
5960:
5955:
5950:
5945:
5940:
5934:
5931:
5930:
5924:
5917:
5916:
5913:
5912:
5910:
5909:
5904:
5899:
5894:
5889:
5883:
5880:
5879:
5876:
5869:
5868:
5865:
5864:
5862:
5861:
5856:
5851:
5846:
5841:
5836:
5831:
5825:
5822:
5821:
5814:
5807:
5806:
5803:
5802:
5800:
5799:
5794:
5789:
5784:
5779:
5774:
5769:
5764:
5759:
5754:
5749:
5744:
5739:
5734:
5728:
5725:
5724:
5718:
5711:
5710:
5704:
5702:
5701:
5694:
5687:
5679:
5673:
5672:
5653:
5634:
5615:
5596:
5589:
5583:
5575:
5574:External links
5572:
5569:
5568:
5542:
5513:
5494:
5473:
5454:
5435:(4): 210â214.
5416:
5387:
5361:
5324:
5275:
5216:
5170:Carl Pomerance
5161:
5154:
5124:
5117:
5101:Carl Pomerance
5083:
5076:
5056:David Bressoud
5040:
4956:
4955:
4953:
4950:
4929:
4916:
4915:
4903:
4900:
4896:
4893:
4887:
4881:
4878:
4872:
4868:
4863:
4859:
4820:
4792:
4789:
4785:
4784:
4753:
4692:Bruckman-Lucas
4688:
4687:
4670:
4669:
4638:
4637:
4628:
4626:
4615:
4611:
4608:
4604:
4601:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4573:
4569:
4522:
4521:
4469:
4455:
4452:
4353:
4350:
4348:is not prime.
4311:
4310:
4301:
4299:
4288:
4284:
4281:
4277:
4274:
4268:
4262:
4259:
4253:
4249:
4246:
4243:
4238:
4234:
4230:
4227:
4224:
4221:
4218:
4214:
4210:
4207:
4204:
4199:
4195:
4191:
4188:
4185:
4182:
4179:
4175:
4156:Therefore, if
4133:
4127:
4124:
4118:
4106:
4105:
4092:
4089:
4085:
4082:
4076:
4070:
4067:
4061:
4057:
4052:
4048:
4044:
4041:
4038:
4035:
4032:
4028:
3991:
3987:
3983:
3980:
3977:
3974:
3971:
3967:
3946:
3924:
3921:
3918:
3914:
3891:
3888:
3885:
3881:
3857:
3854:
3851:
3848:
3776:
3770:
3767:
3761:
3683:
3676:
3669:
3668:
3659:
3657:
3646:
3642:
3639:
3635:
3632:
3627:
3624:
3621:
3616:
3613:
3610:
3606:
3580:
3577:
3574:
3570:
3564:
3561:
3555:
3525:
3522:
3483:
3480:
3477:
3473:
3467:
3464:
3458:
3437:
3434:
3431:
3426:
3422:
3418:
3415:
3391:
3388:
3385:
3304:
3283:
3276:
3231:
3224:
3217:
3210:
3203:
3196:
3189:
3182:
3175:
3171:
3167:
3163:
3159:
3155:
3151:
3147:
3112:
3092:
3072:
3024:
3021:
3018:
3015:
3011:
2990:
2987:
2982:
2979:
2975:
2971:
2966:
2963:
2959:
2955:
2952:
2930:
2927:
2923:
2919:
2914:
2911:
2907:
2903:
2900:
2889:
2888:
2876:
2872:
2868:
2863:
2860:
2856:
2852:
2849:
2846:
2841:
2838:
2834:
2830:
2827:
2824:
2821:
2816:
2813:
2810:
2807:
2803:
2792:
2781:
2777:
2773:
2768:
2765:
2761:
2757:
2752:
2749:
2745:
2741:
2738:
2735:
2732:
2727:
2724:
2721:
2718:
2714:
2699:
2698:
2684:
2678:
2673:
2669:
2665:
2662:
2657:
2652:
2648:
2641:
2636:
2632:
2628:
2625:
2620:
2615:
2611:
2607:
2602:
2599:
2595:
2584:
2571:
2567:
2563:
2558:
2554:
2550:
2545:
2542:
2538:
2514:
2511:
2491:
2480:
2479:
2468:
2465:
2462:
2459:
2454:
2450:
2439:
2428:
2425:
2420:
2416:
2388:
2385:
2382:
2377:
2373:
2369:
2366:
2344:
2341:
2338:
2334:
2311:
2308:
2305:
2301:
2259:John Selfridge
2222:
2218:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2176:
2173:
2170:
2138:
2135:
2131:
2125:
2122:
2116:
2087:
2084:
2081:
2077:
2071:
2067:
2064:
2057:
2052:
2046:
2043:
2037:
2016:
2013:
2010:
2006:
2000:
1997:
1991:
1947:
1944:
1941:
1937:
1931:
1928:
1922:
1898:
1895:
1882:
1879:
1876:
1873:
1870:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1821:
1820:
1808:
1805:
1801:
1798:
1793:
1790:
1783:
1779:
1775:
1772:
1768:
1753:
1752:
1740:
1737:
1733:
1730:
1725:
1722:
1719:
1714:
1710:
1700:
1697:
1693:
1690:
1685:
1682:
1677:
1673:
1609:
1605:
1582:
1578:
1550:
1530:
1527:
1524:
1521:
1518:
1498:
1478:
1458:
1449:(depending on
1438:
1418:
1415:
1412:
1407:
1403:
1382:
1362:
1326:
1325:
1313:
1310:
1306:
1303:
1298:
1295:
1288:
1284:
1280:
1277:
1273:
1258:
1257:
1245:
1242:
1238:
1235:
1230:
1227:
1222:
1218:
1175:
1153:
1149:
1145:
1142:
1133:into the form
1121:
1115:
1112:
1106:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1072:
1069:
1068:
1067:
1034:) for a given
1003:
1002:
991:
987:
984:
980:
977:
972:
969:
964:
960:
935:
929:
926:
920:
893:
892:
881:
877:
874:
870:
867:
862:
859:
856:
853:
848:
844:
827:
808:
802:
799:
793:
772:
765:
758:
751:
656:is composite.
642:
641:
632:
630:
619:
615:
612:
608:
605:
600:
597:
592:
589:
586:
582:
550:
547:
544:
541:
537:
531:
528:
522:
489:
483:
480:
474:
407:
404:
350:
349:
340:
338:
327:
323:
320:
316:
313:
308:
305:
300:
297:
294:
291:
287:
258:
257:
246:
242:
236:
233:
227:
223:
220:
217:
214:
211:
208:
205:
177:
171:
168:
162:
129:
114:
100:
97:
94:
89:
85:
81:
78:
54:
51:
47:Lucas sequence
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7636:
7625:
7622:
7620:
7617:
7616:
7614:
7599:
7595:
7591:
7590:
7587:
7577:
7574:
7573:
7570:
7565:
7560:
7556:
7546:
7543:
7541:
7538:
7537:
7534:
7529:
7524:
7520:
7510:
7507:
7505:
7502:
7501:
7498:
7493:
7488:
7484:
7474:
7471:
7469:
7466:
7465:
7462:
7458:
7452:
7448:
7438:
7435:
7433:
7430:
7428:
7425:
7424:
7421:
7417:
7412:
7408:
7394:
7391:
7390:
7388:
7384:
7378:
7375:
7373:
7370:
7368:
7367:Polydivisible
7365:
7363:
7360:
7358:
7355:
7353:
7350:
7348:
7345:
7344:
7342:
7338:
7332:
7329:
7327:
7324:
7322:
7319:
7317:
7314:
7312:
7309:
7308:
7306:
7303:
7298:
7292:
7289:
7287:
7284:
7282:
7279:
7277:
7274:
7272:
7269:
7267:
7264:
7262:
7259:
7258:
7256:
7253:
7249:
7241:
7238:
7237:
7236:
7233:
7232:
7230:
7227:
7223:
7211:
7208:
7207:
7206:
7203:
7201:
7198:
7196:
7193:
7191:
7188:
7186:
7183:
7181:
7178:
7176:
7173:
7171:
7168:
7166:
7163:
7162:
7160:
7156:
7150:
7147:
7146:
7144:
7140:
7134:
7131:
7129:
7126:
7125:
7123:
7121:Digit product
7119:
7113:
7110:
7108:
7105:
7103:
7100:
7098:
7095:
7094:
7092:
7090:
7086:
7078:
7075:
7073:
7070:
7069:
7068:
7065:
7064:
7062:
7060:
7055:
7051:
7047:
7042:
7037:
7033:
7023:
7020:
7018:
7015:
7013:
7010:
7008:
7005:
7003:
7000:
6998:
6995:
6993:
6990:
6988:
6985:
6983:
6980:
6978:
6975:
6973:
6970:
6968:
6965:
6963:
6960:
6958:
6957:ErdĆsâNicolas
6955:
6953:
6950:
6948:
6945:
6944:
6941:
6936:
6932:
6926:
6922:
6908:
6905:
6903:
6900:
6899:
6897:
6895:
6891:
6885:
6882:
6880:
6877:
6875:
6872:
6870:
6867:
6866:
6864:
6862:
6858:
6852:
6849:
6847:
6844:
6842:
6839:
6837:
6834:
6832:
6829:
6827:
6824:
6823:
6821:
6819:
6815:
6809:
6806:
6804:
6801:
6800:
6798:
6796:
6792:
6786:
6783:
6781:
6778:
6776:
6775:Superabundant
6773:
6771:
6768:
6766:
6763:
6761:
6758:
6756:
6753:
6751:
6748:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6728:
6726:
6723:
6721:
6718:
6716:
6713:
6711:
6708:
6706:
6703:
6701:
6698:
6696:
6693:
6691:
6688:
6686:
6683:
6681:
6678:
6677:
6675:
6673:
6669:
6665:
6661:
6657:
6652:
6648:
6638:
6635:
6633:
6630:
6628:
6625:
6623:
6620:
6618:
6615:
6613:
6610:
6608:
6605:
6603:
6600:
6598:
6595:
6593:
6590:
6588:
6585:
6583:
6580:
6579:
6576:
6572:
6567:
6563:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6534:
6531:
6527:
6522:
6518:
6508:
6505:
6503:
6500:
6498:
6495:
6493:
6490:
6488:
6485:
6483:
6480:
6478:
6475:
6473:
6470:
6468:
6465:
6463:
6460:
6458:
6455:
6453:
6450:
6448:
6445:
6443:
6440:
6438:
6435:
6433:
6430:
6428:
6425:
6423:
6420:
6418:
6415:
6413:
6410:
6409:
6406:
6399:
6395:
6377:
6374:
6372:
6369:
6367:
6364:
6363:
6361:
6357:
6354:
6352:
6351:4-dimensional
6348:
6338:
6335:
6334:
6332:
6330:
6326:
6320:
6317:
6315:
6312:
6310:
6307:
6305:
6302:
6300:
6297:
6295:
6292:
6291:
6289:
6287:
6283:
6277:
6274:
6272:
6269:
6267:
6264:
6262:
6261:Centered cube
6259:
6257:
6254:
6253:
6251:
6249:
6245:
6242:
6240:
6239:3-dimensional
6236:
6226:
6223:
6221:
6218:
6216:
6213:
6211:
6208:
6206:
6203:
6201:
6198:
6196:
6193:
6191:
6188:
6186:
6183:
6181:
6178:
6177:
6175:
6173:
6169:
6163:
6160:
6158:
6155:
6153:
6150:
6148:
6145:
6143:
6140:
6138:
6135:
6133:
6130:
6128:
6125:
6123:
6120:
6119:
6117:
6115:
6111:
6108:
6106:
6105:2-dimensional
6102:
6098:
6094:
6089:
6085:
6075:
6072:
6070:
6067:
6065:
6062:
6060:
6057:
6055:
6052:
6050:
6049:Nonhypotenuse
6047:
6046:
6043:
6036:
6032:
6022:
6019:
6017:
6014:
6012:
6009:
6007:
6004:
6002:
5999:
5998:
5995:
5988:
5984:
5974:
5971:
5969:
5966:
5964:
5961:
5959:
5956:
5954:
5951:
5949:
5946:
5944:
5941:
5939:
5936:
5935:
5932:
5927:
5922:
5918:
5908:
5905:
5903:
5900:
5898:
5895:
5893:
5890:
5888:
5885:
5884:
5881:
5874:
5870:
5860:
5857:
5855:
5852:
5850:
5847:
5845:
5842:
5840:
5837:
5835:
5832:
5830:
5827:
5826:
5823:
5818:
5812:
5808:
5798:
5795:
5793:
5790:
5788:
5787:Perfect power
5785:
5783:
5780:
5778:
5777:Seventh power
5775:
5773:
5770:
5768:
5765:
5763:
5760:
5758:
5755:
5753:
5750:
5748:
5745:
5743:
5740:
5738:
5735:
5733:
5730:
5729:
5726:
5721:
5716:
5712:
5708:
5700:
5695:
5693:
5688:
5686:
5681:
5680:
5677:
5668:
5667:
5662:
5659:
5654:
5649:
5648:
5643:
5640:
5635:
5630:
5629:
5624:
5621:
5616:
5611:
5610:
5605:
5602:
5597:
5594:
5590:
5588:
5584:
5582:
5578:
5577:
5573:
5564:
5560:
5556:
5549:
5547:
5543:
5537:
5532:
5528:
5524:
5517:
5514:
5509:
5505:
5498:
5495:
5483:
5477:
5474:
5469:
5465:
5458:
5455:
5450:
5446:
5442:
5438:
5434:
5430:
5426:
5420:
5417:
5411:
5406:
5402:
5398:
5391:
5388:
5384:(3): 232â242.
5383:
5379:
5372:
5365:
5362:
5357:
5353:
5348:
5343:
5339:
5335:
5328:
5325:
5320:
5316:
5312:
5308:
5303:
5298:
5294:
5290:
5286:
5279:
5276:
5271:
5267:
5262:
5257:
5253:
5249:
5244:
5239:
5235:
5231:
5227:
5220:
5217:
5212:
5208:
5203:
5198:
5194:
5190:
5183:
5180:(July 1980).
5179:
5175:
5171:
5165:
5162:
5157:
5155:0-387-94457-5
5151:
5147:
5143:
5139:
5133:
5131:
5129:
5125:
5120:
5118:0-387-25282-7
5114:
5110:
5106:
5102:
5098:
5092:
5090:
5088:
5084:
5079:
5073:
5068:
5067:
5061:
5057:
5051:
5049:
5047:
5045:
5041:
5036:
5032:
5028:
5024:
5019:
5014:
5010:
5006:
4999:
4995:
4988:
4986:
4984:
4982:
4980:
4978:
4976:
4974:
4972:
4970:
4968:
4966:
4964:
4962:
4958:
4951:
4949:
4947:
4943:
4938:
4936:
4935:Pell sequence
4932:
4925:
4921:
4898:
4894:
4885:
4879:
4876:
4870:
4866:
4861:
4857:
4844:
4843:
4842:
4839:
4835:
4829:
4827:
4826:Pell sequence
4823:
4816:
4812:
4808:
4807:
4802:
4798:
4790:
4788:
4782:
4778:
4774:
4770:
4766:
4762:
4758:
4754:
4752:
4748:
4745:
4744:
4743:
4741:
4737:
4733:
4729:
4728:
4723:
4719:
4714:
4712:
4711:
4706:
4701:
4695:
4693:
4685:
4680:
4675:
4674:
4673:
4667:
4663:
4660:) holds with
4659:
4658:
4653:
4649:
4645:
4644:
4643:
4636:
4629:
4627:
4613:
4606:
4602:
4594:
4591:
4585:
4582:
4579:
4571:
4567:
4559:
4558:
4555:
4553:
4549:
4545:
4540:
4538:
4534:
4530:
4525:
4519:
4514:
4509:
4508:
4507:
4505:
4501:
4498:) holds with
4497:
4496:
4491:
4487:
4482:
4480:
4476:
4472:
4465:
4461:
4453:
4451:
4449:
4445:
4440:
4438:
4434:
4430:
4426:
4422:
4418:
4414:
4410:
4406:
4401:
4399:
4395:
4390:
4388:
4384:
4380:
4376:
4371:
4368:
4366:
4362:
4358:
4351:
4349:
4347:
4343:
4339:
4334:
4332:
4328:
4327:
4322:
4318:
4309:
4302:
4300:
4286:
4279:
4275:
4266:
4260:
4257:
4251:
4247:
4244:
4241:
4236:
4232:
4225:
4222:
4219:
4212:
4208:
4205:
4202:
4197:
4193:
4186:
4183:
4180:
4173:
4165:
4164:
4161:
4159:
4154:
4152:
4148:
4131:
4125:
4122:
4116:
4087:
4083:
4074:
4068:
4065:
4059:
4055:
4050:
4046:
4039:
4036:
4033:
4026:
4018:
4017:
4016:
4014:
4010:
4005:
3989:
3985:
3978:
3975:
3972:
3965:
3944:
3922:
3919:
3916:
3912:
3889:
3886:
3883:
3879:
3869:
3855:
3852:
3849:
3846:
3837:
3835:
3834:
3829:
3825:
3821:
3817:
3816:
3811:
3807:
3803:
3799:
3795:
3791:
3774:
3768:
3765:
3759:
3750:
3746:
3742:
3738:
3734:
3730:
3726:
3721:
3719:
3718:
3713:
3709:
3705:
3701:
3700:
3695:
3694:
3688:
3686:
3679:
3667:
3660:
3658:
3644:
3637:
3633:
3625:
3622:
3619:
3614:
3611:
3608:
3604:
3596:
3595:
3592:
3578:
3575:
3572:
3568:
3562:
3559:
3553:
3544:
3539:
3537:
3533:
3532:
3523:
3521:
3519:
3514:
3509:
3505:
3501:
3497:
3481:
3478:
3475:
3471:
3465:
3462:
3456:
3435:
3432:
3429:
3424:
3420:
3416:
3413:
3405:
3389:
3386:
3383:
3375:
3370:
3368:
3363:
3358:
3354:
3350:
3346:
3341:
3339:
3334:
3332:
3327:
3322:
3318:
3314:
3309:
3307:
3300:
3299:
3294:
3290:
3286:
3279:
3271:
3269:
3265:
3261:
3257:
3253:
3249:
3245:
3241:
3237:
3230:
3223:
3216:
3209:
3202:
3195:
3188:
3181:
3145:
3141:
3138:terms in the
3137:
3134:to determine
3133:
3128:
3126:
3110:
3090:
3070:
3061:
3059:
3055:
3051:
3047:
3044:
3040:
3022:
3019:
3016:
3013:
3009:
2988:
2985:
2980:
2977:
2973:
2969:
2964:
2961:
2957:
2953:
2950:
2928:
2925:
2921:
2917:
2912:
2909:
2905:
2901:
2898:
2874:
2870:
2861:
2858:
2854:
2850:
2847:
2844:
2839:
2836:
2832:
2828:
2825:
2819:
2814:
2811:
2808:
2805:
2801:
2793:
2779:
2775:
2766:
2763:
2759:
2755:
2750:
2747:
2743:
2739:
2736:
2730:
2725:
2722:
2719:
2716:
2712:
2704:
2703:
2702:
2682:
2676:
2671:
2667:
2663:
2660:
2655:
2650:
2646:
2639:
2634:
2630:
2626:
2623:
2618:
2613:
2609:
2605:
2600:
2597:
2593:
2585:
2569:
2565:
2561:
2556:
2552:
2548:
2543:
2540:
2536:
2528:
2527:
2526:
2512:
2509:
2489:
2466:
2463:
2460:
2457:
2452:
2448:
2440:
2426:
2423:
2418:
2414:
2406:
2405:
2404:
2402:
2383:
2380:
2375:
2371:
2364:
2342:
2339:
2336:
2332:
2309:
2306:
2303:
2299:
2290:
2286:
2282:
2277:
2275:
2271:
2267:
2262:
2260:
2256:
2252:
2248:
2244:
2240:
2236:
2220:
2216:
2209:
2206:
2203:
2197:
2194:
2174:
2171:
2168:
2160:
2156:
2152:
2136:
2133:
2129:
2123:
2120:
2114:
2105:
2101:
2085:
2082:
2079:
2075:
2069:
2065:
2062:
2055:
2050:
2044:
2041:
2035:
2014:
2011:
2008:
2004:
1998:
1995:
1989:
1980:
1976:
1972:
1967:
1965:
1961:
1945:
1942:
1939:
1935:
1929:
1926:
1920:
1910:
1908:
1904:
1896:
1894:
1877:
1874:
1871:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1803:
1799:
1791:
1788:
1781:
1777:
1773:
1770:
1766:
1758:
1757:
1756:
1735:
1731:
1723:
1720:
1717:
1712:
1708:
1695:
1691:
1683:
1680:
1675:
1671:
1663:
1662:
1661:
1659:
1655:
1651:
1647:
1642:
1640:
1636:
1635:
1629:
1625:
1607:
1603:
1580:
1576:
1567:
1562:
1548:
1528:
1525:
1522:
1519:
1516:
1496:
1476:
1456:
1436:
1416:
1413:
1410:
1405:
1401:
1380:
1360:
1351:
1349:
1348:
1343:
1339:
1335:
1331:
1308:
1304:
1296:
1293:
1286:
1282:
1278:
1275:
1271:
1263:
1262:
1261:
1240:
1236:
1228:
1225:
1220:
1216:
1208:
1207:
1206:
1204:
1200:
1196:
1193:for a given (
1192:
1187:
1173:
1151:
1147:
1143:
1140:
1119:
1113:
1110:
1104:
1100:
1097:
1094:
1088:
1082:
1070:
1064:
1063:
1062:
1060:
1056:
1051:
1049:
1045:
1041:
1037:
1033:
1032:
1027:
1022:
1020:
1016:
1012:
1008:
989:
982:
978:
970:
967:
962:
958:
950:
949:
948:
933:
927:
924:
918:
909:
904:
902:
898:
879:
872:
868:
860:
857:
854:
851:
846:
842:
834:
833:
832:
830:
823:
806:
800:
797:
791:
782:
777:
775:
768:
761:
754:
746:
742:
737:
733:
729:
725:
722:Examples: If
720:
718:
714:
710:
706:
702:
698:
694:
690:
686:
682:
678:
677:
672:
668:
664:
663:
657:
655:
651:
650:
640:
633:
631:
617:
610:
606:
598:
595:
590:
587:
584:
580:
572:
571:
568:
566:
565:
548:
545:
542:
539:
535:
529:
526:
520:
510:
508:
504:
487:
481:
478:
472:
463:
458:
456:
455:
450:
446:
442:
439:for a given (
438:
433:
431:
430:
425:
421:
417:
414:for a given (
413:
405:
403:
401:
396:
394:
390:
389:
383:
381:
377:
373:
369:
365:
361:
357:
348:
341:
339:
325:
318:
314:
306:
303:
295:
289:
285:
277:
276:
273:
271:
267:
263:
244:
240:
234:
231:
225:
221:
218:
215:
209:
203:
196:
195:
194:
192:
191:Jacobi symbol
175:
169:
166:
160:
151:
146:
144:
140:
136:
132:
125:
121:
117:
98:
95:
92:
87:
83:
79:
76:
68:
64:
60:
52:
50:
48:
44:
40:
36:
32:
19:
7624:Pseudoprimes
7331:Transposable
7195:Narcissistic
7102:Digital root
7022:Super-Poulet
6982:JordanâPĂłlya
6931:prime factor
6836:Noncototient
6803:Almost prime
6785:Superperfect
6760:Refactorable
6755:Quasiperfect
6730:Hyperperfect
6616:
6571:Pseudoprimes
6542:WallâSunâSun
6477:Ordered Bell
6447:FussâCatalan
6359:non-centered
6309:Dodecahedral
6286:non-centered
6172:non-centered
6074:Wolstenholme
5819:× 2 ± 1
5816:
5815:Of the form
5782:Eighth power
5762:Fourth power
5664:
5645:
5626:
5607:
5554:
5526:
5522:
5516:
5507:
5503:
5497:
5485:. Retrieved
5476:
5467:
5463:
5457:
5432:
5428:
5419:
5400:
5396:
5390:
5381:
5377:
5364:
5337:
5333:
5327:
5292:
5288:
5278:
5233:
5229:
5219:
5192:
5188:
5164:
5141:
5104:
5065:
5008:
5004:
4945:
4941:
4939:
4927:
4923:
4919:
4917:
4830:
4818:
4814:
4810:
4804:
4800:
4796:
4794:
4786:
4780:
4776:
4772:
4768:
4764:
4760:
4756:
4746:
4739:
4735:
4731:
4730:) holds for
4725:
4721:
4717:
4715:
4708:
4696:
4691:
4689:
4671:
4665:
4661:
4655:
4651:
4647:
4641:
4630:
4551:
4547:
4543:
4541:
4536:
4528:
4526:
4523:
4503:
4499:
4493:
4489:
4485:
4483:
4478:
4474:
4467:
4463:
4459:
4457:
4441:
4432:
4428:
4424:
4416:
4412:
4408:
4404:
4402:
4397:
4393:
4391:
4386:
4382:
4378:
4374:
4372:
4369:
4364:
4356:
4355:
4345:
4341:
4337:
4335:
4324:
4320:
4316:
4314:
4303:
4157:
4155:
4150:
4107:
4008:
4006:
3871:Recall that
3870:
3838:
3831:
3827:
3823:
3819:
3813:
3809:
3805:
3801:
3793:
3789:
3748:
3744:
3740:
3736:
3732:
3728:
3724:
3722:
3715:
3711:
3707:
3703:
3697:
3691:
3689:
3681:
3674:
3672:
3661:
3542:
3540:
3535:
3529:
3527:
3508:extra strong
3507:
3503:
3499:
3495:
3403:
3374:extra strong
3373:
3371:
3356:
3352:
3348:
3344:
3342:
3337:
3335:
3320:
3316:
3312:
3310:
3302:
3296:
3292:
3288:
3281:
3274:
3272:
3267:
3263:
3259:
3255:
3251:
3247:
3243:
3239:
3235:
3228:
3221:
3214:
3207:
3200:
3193:
3186:
3179:
3174:= 22, 101100
3143:
3139:
3135:
3131:
3129:
3124:
3062:
3057:
3053:
3049:
3045:
3038:
2890:
2700:
2481:
2288:
2284:
2280:
2278:
2273:
2269:
2265:
2263:
2254:
2250:
2246:
2242:
2238:
2234:
2158:
2154:
2150:
2103:
2099:
2027:. Note that
1978:
1974:
1970:
1968:
1963:
1959:
1958:, so that ÎŽ(
1911:
1902:
1900:
1822:
1754:
1657:
1653:
1649:
1645:
1643:
1638:
1633:
1631:
1627:
1623:
1565:
1563:
1352:
1345:
1341:
1337:
1333:
1329:
1327:
1259:
1202:
1198:
1194:
1190:
1188:
1075:Now, factor
1074:
1058:
1054:
1052:
1047:
1043:
1039:
1035:
1029:
1024:We will see
1023:
1021:= −1.
1018:
1014:
1010:
1006:
1004:
907:
905:
900:
896:
894:
825:
821:
780:
778:
770:
763:
756:
749:
735:
731:
727:
723:
721:
716:
712:
708:
704:
700:
696:
692:
688:
684:
680:
674:
670:
666:
660:
658:
653:
647:
645:
634:
562:
511:
461:
459:
452:
448:
444:
440:
436:
434:
427:
423:
419:
415:
411:
409:
397:
386:
384:
375:
371:
367:
363:
359:
355:
353:
342:
269:
261:
259:
193:. We define
149:
147:
138:
134:
127:
123:
119:
112:
66:
62:
58:
56:
34:
30:
29:
7352:Extravagant
7347:Equidigital
7302:permutation
7261:Palindromic
7235:Automorphic
7133:Sum-product
7112:Sum-product
7067:Persistence
6962:ErdĆsâWoods
6884:Untouchable
6765:Semiperfect
6715:Hemiperfect
6376:Tesseractic
6314:Icosahedral
6294:Tetrahedral
6225:Dodecagonal
5926:Recursively
5797:Prime power
5772:Sixth power
5767:Fifth power
5747:Power of 10
5705:Classes of
5529:: 173â177.
5403:: 347â354.
4668:= −1.
4421:twin primes
4392:Therefore,
3836:) is true.
3402:. Then try
3170:= 11, 10110
2399:steps; see
1568:= â1, then
1564:We can set
1007:pseudoprime
779:First, let
776:= 10, etc.
400:Mathematica
358:hold, then
69:> 0 and
7613:Categories
7564:Graphemics
7437:Pernicious
7291:Undulating
7266:Pandigital
7240:Trimorphic
6841:Nontotient
6690:Arithmetic
6304:Octahedral
6205:Heptagonal
6195:Pentagonal
6180:Triangular
6021:SierpiĆski
5943:Jacobsthal
5742:Power of 3
5737:Power of 2
5510:: 155â157.
5302:2006.14425
5243:1903.06820
5060:Stan Wagon
4952:References
4423:. Such an
3957:, namely,
3166:= 10, 1011
1050:sequence.
1009:for this (
855:6616217487
719:is prime.
567:) becomes
418:) pair is
366:prime. If
7321:Parasitic
7170:Factorion
7097:Digit sum
7089:Digit sum
6907:Fortunate
6894:Primorial
6808:Semiprime
6745:Practical
6710:Descartes
6705:Deficient
6695:Betrothed
6537:Wieferich
6366:Pentatope
6329:pyramidal
6220:Decagonal
6215:Nonagonal
6210:Octagonal
6200:Hexagonal
6059:Practical
6006:Congruent
5938:Fibonacci
5902:Loeschian
5666:MathWorld
5647:MathWorld
5628:MathWorld
5609:MathWorld
5531:CiteSeerX
5405:CiteSeerX
5342:CiteSeerX
5319:220055722
4867:≡
4779:dividing
4592:≡
4248:⋅
4242:≡
4223:−
4209:⋅
4203:≡
4056:≡
4037:−
3853:±
3850:≠
3796:= 5 (see
3620:≡
3576:−
3479:−
3430:−
3162:= 5, 1010
2954:⋅
2902:⋅
2851:⋅
2829:⋅
2740:⋅
2624:−
2562:⋅
2381:
2207:−
2161:, we set
2083:−
2063:−
2012:−
1943:−
1846:−
1834:≤
1823:for some
1789:≡
1774:⋅
1721:±
1718:≡
1681:≡
1526:
1520:⋅
1411:−
1294:≡
1279:⋅
1226:≡
1201:with GCD(
1144:⋅
1101:−
1083:δ
968:≡
858:≡
596:≡
543:−
445:composite
372:composite
304:≡
290:δ
222:−
204:δ
93:−
39:composite
7393:Friedman
7326:Primeval
7271:Repdigit
7228:-related
7175:Kaprekar
7149:Meertens
7072:Additive
7059:dynamics
6967:Friendly
6879:Sociable
6869:Amicable
6680:Abundant
6660:dynamics
6482:Schröder
6472:Narayana
6442:Eulerian
6432:Delannoy
6427:Dedekind
6248:centered
6114:centered
6001:Amenable
5958:Narayana
5948:Leonardo
5844:Mersenne
5792:Powerful
5732:Achilles
5140:(1996).
5103:(2005).
5062:(2000).
4813:= 2 and
4771:−
4767:+ 1) | (
4759:+ 1) | (
4664:= 1 and
4502:= 1 and
4462:= 1 and
3747:so that
3158:= 4, 101
3154:= 2, 100
1656:) where
1186:is odd.
1038:, we do
824:) = 20,
689:probable
447:integer
65:, where
7566:related
7530:related
7494:related
7492:Sorting
7377:Vampire
7362:Harshad
7304:related
7276:Repunit
7190:Lychrel
7165:Dudeney
7017:StĂžrmer
7012:Sphenic
6997:Regular
6935:divisor
6874:Perfect
6770:Sublime
6740:Perfect
6467:Motzkin
6422:Catalan
5963:Padovan
5897:Leyland
5892:Idoneal
5887:Hilbert
5859:Woodall
5449:2689212
5270:1680879
5248:Bibcode
5211:2006210
5035:0583518
5027:2006406
4933:as the
4838:A099011
4836::
4703:in the
4700:A141137
4682:in the
4679:A005845
4516:in the
4513:A081264
4145:is the
4108:(Here,
3516:in the
3513:A217719
3365:in the
3362:A217255
3329:in the
3326:A217120
3291:, (mod
3150:= 1, 10
745:A006190
743::
376:usually
189:be the
7432:Odious
7357:Frugal
7311:Cyclic
7300:Digit-
7007:Smooth
6992:Pronic
6952:Cyclic
6929:Other
6902:Euclid
6552:Wilson
6526:Primes
6185:Square
6054:Polite
6016:Riesel
6011:Knödel
5973:Perrin
5854:Thabit
5839:Fermat
5829:Cullen
5752:Square
5720:Powers
5533:
5447:
5407:
5344:
5317:
5268:
5209:
5152:
5115:
5074:
5033:
5025:
4918:with (
4853:
4431:+1 = (
3826:, and
3731:, and
3696:) or (
3502:, and
3357:strong
3351:, and
3338:strong
3319:, and
3287:, and
3266:, and
3227:, and
3123:, mod
3043:modulo
2287:, and
2279:Given
2253:, and
2098:. (If
1969:Given
1893:pair.
1541:, for
1342:strong
1166:where
685:likely
126:) and
111:, let
43:primes
7473:Prime
7468:Lucky
7457:sieve
7386:Other
7372:Smith
7252:Digit
7210:Happy
7185:Keith
7158:Other
7002:Rough
6972:Giuga
6437:Euler
6299:Cubic
5953:Lucas
5849:Proth
5487:5 May
5445:JSTOR
5374:(PDF)
5315:S2CID
5297:arXiv
5238:arXiv
5207:JSTOR
5185:(PDF)
5023:JSTOR
5001:(PDF)
4749:is a
4458:When
4149:; if
3343:When
3311:When
3136:which
1966:+ 1.
1332:<
1026:below
1017:= 3,
769:= 3,
762:= 1,
755:= 0,
738:s is
726:= 3,
695:that
693:prove
266:prime
264:is a
7427:Evil
7107:Self
7057:and
6947:Blum
6658:and
6462:Lobb
6417:Cake
6412:Bell
6162:Star
6069:Ulam
5968:Pell
5757:Cube
5489:2019
5150:ISBN
5113:ISBN
5072:ISBN
4834:OEIS
4705:OEIS
4684:OEIS
4518:OEIS
4321:n, Q
3810:only
3804:and
3751:and
3743:and
3708:both
3518:OEIS
3367:OEIS
3336:The
3331:OEIS
2324:and
2187:and
2102:and
1962:) =
1840:<
1622:are
1595:and
1469:and
1373:and
1338:P, Q
1203:n, D
1195:P, Q
1053:Let
1011:P, Q
897:P, Q
741:OEIS
707:and
441:P, Q
416:P, Q
148:Let
61:and
37:are
33:and
7545:Ban
6933:or
6452:Lah
5559:doi
5437:doi
5352:doi
5307:doi
5256:doi
5197:doi
5013:doi
4895:mod
4713:).
4603:mod
4542:If
4527:If
4342:any
4276:mod
4084:mod
4007:If
3839:If
3720:).
3684:n+1
3677:n+1
3634:mod
3541:If
3305:n+1
3284:n+1
3277:n+1
2891:If
2502:to
2372:log
2357:in
2241:or
1800:mod
1755:or
1732:mod
1692:mod
1644:An
1523:log
1350:).
1305:mod
1260:or
1237:mod
1066:...
1040:not
983:119
979:mod
963:120
928:119
901:not
869:mod
683:is
607:mod
512:If
420:any
370:is
364:not
362:is
356:not
315:mod
260:If
7615::
5663:.
5644:.
5625:.
5606:.
5545:^
5527:31
5525:.
5508:32
5506:.
5466:.
5443:.
5433:47
5431:.
5401:28
5399:.
5382:27
5380:.
5376:.
5350:.
5338:66
5336:.
5313:.
5305:.
5293:90
5291:.
5266:MR
5264:.
5254:.
5246:.
5234:70
5232:.
5228:.
5205:.
5193:35
5191:.
5187:.
5176:;
5172:;
5148:.
5144:.
5127:^
5111:.
5099:;
5086:^
5058:;
5043:^
5031:MR
5029:.
5021:.
5009:35
5007:.
5003:.
4960:^
4944:,
4922:,
4795:A
4755:2(
4716:A
4686:),
4646:a
4550:,
4520:).
4484:A
4450:.
4411:=
4333:.
4015:,
4004:.
3822:,
3792:=
3727:,
3687:.
3520:)
3498:,
3369:)
3347:,
3333:)
3315:,
3308:.
3280:,
3270:.
3262:,
3258:,
3254:,
3250:,
3246:,
3242:,
3232:44
3225:22
3220:,
3218:11
3213:,
3211:10
3206:,
3199:,
3192:,
3185:,
3127:.
3083:,
3060:.
2283:,
2261:.
2249:,
1652:,
1189:A
925:13
873:19
847:20
828:20
801:19
798:13
748::
736:U'
703:,
435:A
410:A
382:.
145:.
137:,
122:,
49:.
5817:a
5698:e
5691:t
5684:v
5669:.
5650:.
5631:.
5612:.
5565:.
5561::
5539:.
5491:.
5468:4
5451:.
5439::
5413:.
5358:.
5354::
5321:.
5309::
5299::
5272:.
5258::
5250::
5240::
5213:.
5199::
5158:.
5121:.
5080:.
5037:.
5015::
4946:Q
4942:P
4930:n
4928:U
4924:Q
4920:P
4902:)
4899:n
4892:(
4886:)
4880:n
4877:2
4871:(
4862:n
4858:U
4821:n
4819:U
4815:Q
4811:P
4806:1
4801:n
4783:.
4781:n
4777:p
4773:p
4769:n
4765:p
4761:n
4757:p
4747:n
4740:n
4736:P
4732:Q
4727:5
4722:n
4710:1
4666:Q
4662:P
4657:5
4652:n
4635:)
4633:5
4631:(
4614:.
4610:)
4607:n
4600:(
4595:P
4589:)
4586:Q
4583:,
4580:P
4577:(
4572:n
4568:V
4552:Q
4548:n
4544:n
4537:n
4529:n
4504:Q
4500:P
4495:1
4490:n
4479:Q
4477:,
4475:P
4473:(
4470:n
4468:U
4464:Q
4460:P
4433:p
4429:n
4425:n
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4415:(
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4317:n
4308:)
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4304:(
4287:.
4283:)
4280:n
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4252:(
4245:Q
4237:2
4233:/
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4220:n
4217:(
4213:Q
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4194:/
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4104:.
4091:)
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4075:)
4069:n
4066:Q
4060:(
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4047:/
4043:)
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4034:n
4031:(
4027:Q
4009:n
3990:2
3986:/
3982:)
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3973:n
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3966:Q
3945:Q
3923:1
3920:+
3917:n
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3760:(
3749:D
3745:Q
3741:P
3737:Q
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3717:2
3712:n
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3693:2
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3664:3
3662:(
3645:.
3641:)
3638:n
3631:(
3626:Q
3623:2
3615:1
3612:+
3609:n
3605:V
3579:1
3573:=
3569:)
3563:n
3560:D
3554:(
3543:n
3536:n
3531:2
3504:Q
3500:P
3496:D
3482:1
3476:=
3472:)
3466:n
3463:D
3457:(
3436:Q
3433:4
3425:2
3421:P
3417:=
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3404:P
3390:1
3387:=
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3353:Q
3349:P
3345:D
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3317:P
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3298:2
3293:n
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3260:Q
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3020:+
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2887:.
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2871:/
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2823:(
2820:=
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2809:k
2806:2
2802:V
2780:2
2776:/
2772:)
2767:k
2764:2
2760:V
2756:+
2751:k
2748:2
2744:U
2737:P
2734:(
2731:=
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2720:k
2717:2
2713:U
2697:.
2683:2
2677:2
2672:k
2668:U
2664:D
2661:+
2656:2
2651:k
2647:V
2640:=
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2614:k
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2606:=
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2598:2
2594:V
2570:k
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2557:k
2553:U
2549:=
2544:k
2541:2
2537:U
2513:k
2510:2
2490:k
2467:1
2464:=
2461:P
2458:=
2453:1
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2427:1
2424:=
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2415:U
2387:)
2384:n
2376:2
2368:(
2365:O
2343:1
2340:+
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2243:Q
2239:P
2235:n
2221:4
2217:/
2213:)
2210:D
2204:1
2201:(
2198:=
2195:Q
2175:1
2172:=
2169:P
2159:D
2155:D
2151:D
2137:0
2134:=
2130:)
2124:n
2121:D
2115:(
2104:n
2100:D
2086:1
2080:=
2076:)
2070:n
2066:k
2056:(
2051:)
2045:n
2042:k
2036:(
2015:1
2009:=
2005:)
1999:n
1996:D
1990:(
1979:D
1975:D
1971:n
1964:n
1960:n
1946:1
1940:=
1936:)
1930:n
1927:D
1921:(
1903:n
1881:)
1878:Q
1875:,
1872:P
1869:(
1849:1
1843:s
1837:r
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1807:)
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1797:(
1792:0
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1739:)
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1713:d
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1699:)
1696:n
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1549:x
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1517:c
1497:x
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1457:P
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1302:(
1297:0
1287:r
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1276:d
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1244:)
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1234:(
1229:0
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1217:U
1199:n
1174:d
1152:s
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1141:d
1120:)
1114:n
1111:D
1105:(
1098:n
1095:=
1092:)
1089:n
1086:(
1059:P
1055:Q
1048:U
1044:n
1036:n
1031:2
1019:Q
1015:P
990:.
986:)
976:(
971:0
959:U
934:)
919:(
908:n
880:.
876:)
866:(
861:0
852:=
843:U
826:U
822:n
807:)
792:(
781:n
773:3
771:U
766:2
764:U
759:1
757:U
752:0
750:U
732:D
728:Q
724:P
717:n
713:n
709:Q
705:P
701:D
697:n
681:n
676:2
671:n
667:n
662:2
654:n
649:2
639:)
637:2
635:(
618:.
614:)
611:n
604:(
599:0
591:1
588:+
585:n
581:U
564:1
549:,
546:1
540:=
536:)
530:n
527:D
521:(
488:)
482:n
479:D
473:(
462:D
454:1
449:n
429:1
424:n
388:1
368:n
360:n
347:)
345:1
343:(
326:.
322:)
319:n
312:(
307:0
299:)
296:n
293:(
286:U
270:Q
262:n
245:.
241:)
235:n
232:D
226:(
219:n
216:=
213:)
210:n
207:(
176:)
170:n
167:D
161:(
150:n
139:Q
135:P
133:(
130:k
128:V
124:Q
120:P
118:(
115:k
113:U
99:Q
96:4
88:2
84:P
80:=
77:D
67:P
63:Q
59:P
20:)
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