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THE JOY OF SEQS!
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The multiplicative order of 2 modulus q.
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Consider any odd positive integer q > 1. Write the following sequence of
integers clockwise in a circle:- 1,2,3,4,5,......,q.
Starting at 1, move 1 place clockwise. Position 2 is arrived at. Continuing,
move 2 places clockwise. Position 4 is arrived at. Keep going clockwise in
steps determined by the initial position. Eventually, one arrives back at
position 1. Position q can never be arrived at.
Examples, where s is the sum of all the positions used and S is the sum of
all the positions:-
1:- q = 5, the positions are 1,2,4,3,1. So 4 different positions are used,
where q = 4.1 + 1, s = 10 = 2q, S = 15 = 3q.
2:- q = 11, the positions are 1,2,4,8,5,10,9,7,3,6,1. So 10 positions are used,
where q = 10.1 + 1, s = 55 = 5q, S = 66 = 6q.
3:- q = 17, the positions are 1,2,4,8,16,15,13,9,1. So 8 positions are used,
where q = 8.2 + 1, s = 68 = 4q, S = 153 = 9q.
4:- q = 43, the positions are 1,2,4,8,16,32,21,42,41,39,35,27,11,22,1. So 14
positions are used, where q = 14.3 + 1, s = 301 = 7q, S = 946 = 22q.
In general, for any odd positive integer q, if the number of positions used
is denoted by m, then q = m.n + 1 for some positive integer n. Notice that
in each case, (S-q)/s = n = (q-1)/m, from which we obtain
q = (m.S + s)/(m + s). Check:- m = 14, s = 301, S = 946,
so q = (14.946 + 301)/(14 + 301) = 13545/315 = 43.
Put another way, m = (s(q-1))/(S-q), so m = (301.42)/903 = 14.
The number m is the multiplicative order (m.o.) of 2 modulus the odd number q.
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Another way of saying this is as follows:-
The multiplicative order (m.o.) of 2 modulus q is the SMALLEST POWER m of 2
such that 2^m = 1 (mod q), i.e. the SMALLEST POWER m to which 2 must be
raised to leave a remainder of 1 when divided by q. Re-using the above examples:-
1:- The m.o. of order 2 mod 5 is 4, because
2^1 = 2 (mod5), 2^2 = 4 (mod5), 2^3 = 3 (mod5), 2^4 = 1 (mod5),
where 5 = 4.1 + 1. So for q = 5, m = 4 and n = 1.
2:- The m.o. of order 2 mod 11 is 10, because
2^1 = 2 (mod11), 2^2 = 4 (mod11), 2^3 = 8 (mod11), 2^4 = 5 (mod11),
2^5 = 10 (mod11), 2^6 = 9 (mod11), 2^7 = 7 (mod11), 2^8 = 3 (mod11),
2^19 = 6 (mod11), 2^10 = 1 (mod11),
where 11 = 10.1 + 1. So for q = 11, m = 10 and n = 1.
3:- The m.o. of order 2 mod 17 is 8, because
2^1 = 2 (mod17), 2^2 = 4 (mod17), 2^3 = 8 (mod17), 2^4 = 16 (mod17),
2^5 = 15 (mod17), 2^6 = 13 (mod17), 2^7 = 9 (mod17), 2^8 = 1 (mod17),
where 17 = 8.2 + 1. So for q = 17, m = 8 and n = 2.
4:- The m.o. of order 2 mod 43 is 14, because
2^1 = 2 (mod43), 2^2 = 4 (mod43), 2^3 = 8 (mod43), 2^4 = 16 (mod43),
2^5 = 32 (mod43), 2^6 = 21 (mod43), 2^7 = 42 (mod43), 2^8 = 41 (mod43),
2^9 = 39 (mod43), 2^10 = 35 (mod43), 2^11 = 27 (mod43), 2^12 = 11 (mod43),
2^13 = 22 (mod43), 2^14 = 1 (mod43),
where 43 = 14.3 + 1. So for q = 43, m = 14 and n = 3.
I like to think of m as the wavelength and n as the frequency of 2 mod q.
These are further examples:-
1:- The m.o. of 2 modulo 7 is 3, because
2^1 = 2 (mod 7), 2^2 = 4 (mod 7), 2^3 = 8 = 1 (mod 7),
and m = 3 for p = 7 is the SMALLEST POWER with this property.
p = 7, m = 3, n = 2.
2:- m of 2 mod 4051 is 50. 2^((4051-1)/81) = 2^50 = 1 (4051).
p = 4051, m = 50, n = 81.
3:- m of 2 mod 8101 is 100. 2^((8101-1)/81) = 2^100 = 1 (8101).
p = 8101, m = 100, n = 81.
4:- m of 2 mod 28001 is 500. 2^((28001-1)/56) = 2^500 = 1 (28001).
p = 28001, m = 500, n = 56.
5:- m of 2 mod 51001 is 1020. 2^((51001-1)/50) = 2^1020 = 1 (51001).
p = 51001, m = 1020, n = 50.
6:- m of 2 mod 55001 is 2750. 2^((55001-1)/20) = 2^2750 = 1 (55001).
p = 55001, m = 2750, n = 20.
7:- m of 2 mod 69001 is 3450. 2^((69001-1)/20) = 2^3450 = 1 (69001).
p = 69001, m = 3450, n = 20.
8:- m of 2 mod 81001 is 2250. 2^((81001-1)/40) = 2^2250 = 1 (81001).
p = 81001, m= 2250, n = 40.
The odd integers q can be subdivided into the odd primes denoted by p,
the pseudoprimes of m.o. 2 denoted by P, and other odd composite numbers.
Such pseudoprimes are also called Poulet numbers.
From the equation q = m.n + 1, it follows that primes p such that the m.o.
m of 2 modulus p is (p-1)/n, for some for positive integer n, i.e. m = (p-1)/n.
Then for each n, the following tables list the corresponding primes p of
m.o. 2 modulus p, i.e. 2^((p-1)/n) = 1 (mod p).
These tables also list the pseudoprimes P of m.o. 2 modulus P,
i.e. 2^((P-1)/n) = 1 (mod P).
The Poulet numbers are composite numbers which behave like primes and
as such are not filtered out by the QBASIC algorthm employed in the search.
P = 341 = 11.31, m = 10, n = 34 is the smallest example,
i.e. 2^((341-1)/34 = 2^10 = 1024 = 1023 + 1 = (3.11.31) + 1 =
(3.341) + 1 = 1 (mod 341).
P = 561 = 3.11.17 , m = 40 , n = 14 is the next smallest example,
i.e. 2^((561-1)/14 = 2^40 = 1099511627776 = 1099511627775 + 1 =
(3.5.5.11.17.31.41.61681) + 1 = (1959913775.561) + 1 = 1 (mod 561).
n q = p or P (given is parentheses)
1:- 3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149,
163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379,
389, 419, 421, 443, 461, 467, 491, 509, 523, 541, 547, 557, 563, 587,
613, 619, 653, 659, 661, 677, 701, 709, 757, 773, 787, 797, 821, 827,
829, 853, 859, 877, 883, 907, 941, 947, 1019, 1061, 1091, 1109, 1117.
2^((3-1)/1) = 1 (3), 2^((5-1)/1) = 1 (5).
2:- 7, 17, 23, 41, 47, 71, 79, 97, 103, 137, 167, 191, 193, 199, 239, 263.
271, 311, 313, 359, 367, 383, 401, 409, 449, 463, 479, 487, 503, 521,
569, 599, 607, 647, 719, 743, 751, 761, 769, 809, 823, 839, 857, 863,
887, 929, 967, 977, 983, 991, 1009, 1031, 1039, 1063, 1087, 1129, 1151.
2^((7-1)/2) = 1 (7), 2^((17-1)/2) = 1 (17).
3:- 43, 109, 157, 229, 277, 283, 307, 499, 643, 691, 733, 739, 811, 997,
1021, 1051, 1069, 1093, 1459, 1579, 1597, 1627, 1699, 1723, 1789, 1933,
2179, 2203, 2251, 2341, 2347, 2749, 2917, 3163, 3181, 3229, 3259, 3373,
4027, 4339, 4549, 4597, 4651, 4909, 5101, 5197, 5323, 5413, 5437, 5653.
2^((43-1)/3) = 1 (43), 2^((109-1)/3) = 1 (109).
4:- 113, 281, 353, 577, 593, 617, 1033, 1049, 1097, 1153, 1193, 1201, 1481.
1601, 1889, 2129, 2273, 2393, 2473, 3049, 3089, 3137, 3217, 3313, 3529,
3673, 3833, 4001, 4217, 4289, 4457, 4801, 4817, 4937, 5233, 5393, 5881,
6121, 6521, 6569, 6761, 6793, 6841, 7129, 7481, 7577, 7793, 7817, 7841.
2^((113-1)/4) = 1 (113), 2^((281-1)/4) = 1 (281).
5:- 251, 571, 971, 1181, 1811, 2011, 2381, 2411, 3221, 3251, 3301, 3821,
4211, 4861, 4931, 5021, 5381, 5861, 6221, 6571, 6581, 8461, 8501, 9091,
9461, 10061, 10211, 10781, 11251, 11701, 11941, 12541, 13171, 13381,
13421, 13781, 14251, 15541, 16091, 16141, 16451, 16661, 16691, 16811.
6:- 31, 223, 433, 439, 457, 727, 919, 1327, 1399, 1423, 1471, 1831, 1999,
2017, 2287, 2383, 2671, 2767, 2791, 2953, 3271, 3343, 3457, 3463, 3607,
3631, 3823, 3889, 4129, 4423, 4519, 4567, 4663, 4729, 4759, 5167, 5449,
5503, 5953, 6007, 6079, 6151, 6217, 6271, 6673, 6961, 6967, 7321, 7369.
7:- 1163, 1709, 2003, 3109, 3389, 3739, 5237, 5531, 5867, 7309, 9157, 9829,
10627, 10739, 11117, 11243, 11299, 11411, 11467, 13259, 18803, 20147,
20483, 21323, 21757, 27749, 27763, 29947, 30773, 31123, 31627, 32803,
33461, 33587, 34469, 35323, 35771, 37339, 44843, 45179, 45557, 46523.
8:- 73, 89, 233, 937, 1217, 1249, 1289, 1433, 1553, 1609, 1721, 1913, 2441,
2969, 3257, 3449, 4049, 4201, 4273, 4297, 4409, 4481, 4993, 5081, 5297,
5689, 6089, 6449, 6481, 6689, 6857, 7121, 7529, 7993, 8081, 8609, 8969,
9137, 9281, 9769, 10337, 10369, 10433, 10937, 11177, 11257, 11617, 11633,
11657, 11801, 12041, 12073, 12409, 12457, 12577, 13049, 13337, 13417,
13441, 13633, 14321, 14537, 14633, 14753, 14969, 15017, 15569, 16249,
16361, 16417, 16433, 16633, 16673, 17137, 17209, 17737, 17881, 18257,
18433, 18481, 18521, 18617, 19289, 19793, 20113, 20297, 20353, 20521,
20809, 20921, 21017, 21433, 21673, 21977, 22697, 23057, 23369, 23801.
9:- 397, 7867, 10243, 10333, 12853, 13789, 14149, 14293, 14563, 15643,
17659, 18379, 18541, 21277, 21997, 23059, 23203, 26731, 27739, 29179,
29683, (30889), 31771, 34147, 35461, 35803, 36541, 37747, 39979, 40213,
40429, 41131, 41491, 44029, 44101, 44587, 49339, 55117, 56179, 56611,
58771, 59707, 60139, 61291, 64171, 65053, 65557, 65701, 69859, 71443
72883, 75781, 76123, 78157, 78283, 78301, 79579, 79669, 79939, 81667,
82963. 86077, 87013, (88561), 88813, 92269, 92683, 93133, 93187, 93997.
10:- 151, 241, 431, 641, 911, 3881, 4751, 4871, 5441, 5471, 5641, 5711, (6601).
6791,6871, 8831, 9041, 9431, 10711, 12721, 13751, 14071, 14431, 14591,
15551, 16631, 16871, 17231, 17681, 17791, 18401, 19031, 19471, 21401,
25111, 25391, 25561, 26921, 27031, 27631, 28351, 31391, 32401, 32801,
32911, 33191, 34031, 35111, 36151, 37871, 38671, 38921, 40351, 45751,
47911, 48271, 48481, 48991, 49031, 51071, 51151, 51631, 51991, 52631.
11:- 331, 1013, 4643, 12101, 12893, 16061, 17117, 23893, 25763, 25939, 28403,
30493, 32429, 32957, 34739, 36389, 38149, 39139, 42043, 44771, 45541,
46861, 53923, 57773, 59621, 60611, 81533, 85229, 87187, 89123, 92357.
12:- 1753, 2281, 2689, 4153, 5113, 5569, 9721, 14953, 19417, 19777, 21193,
21529, 21577, 22441, 23473, 23977, 26209, 26497, 27673, 28537, 29017,
30937, 33577, 33937, 35401, 35809, 36721, 37057, 38713, 41233, 42937,
43753, 43777, 43969, 45481, 47569, 48673, 49057, 50593, 52321, 57193.
13:- 4421, 4733, 5669, 5981, 8581, 9413, 10453, 11597, 13963, 19267, 22699,
22907, 23557, 25117, 25819, 28627, 32579, 35491, 41549, 44773, 44851,
46619, 48179, 52859, 64403, 67757, 69317, 69499, 70019, 73243, 79691.
14:- (561), 631, 673, 953, 4271, 4999, 5279, 6959, 7001, 7351, 13007, 15233,
15583, 16073, 18439, 20063, 20399, 20441, 22303, 26111, 31081, 33391,
33503, 34847, 36527, 37199, 38711, 42743, 45263, 45319, 45461, 46649,
48119, 48623, 49169, 50849, 51647, 54167, 55343, 57527, 58913, 59921.
15:- 3061, 3331, 3541, 5821, 9781, 17491, 18061, 22291, 24421, 27541, 28621,
32251, 35221, 42331, 43891, 44971, 47221, 48091, 52501, 55021, 55171,
57301, 61141, 67021, 67651, 68581, 70051, 70891, 71341, 82021, 84811.
16:- 257, 337, 881, 2657, 4721, 6353, 7489, 10177, 12049, 14897, 18593,
19249, 21713, 25121, 25409, 25537, 25793, 27809, 28657, 28817, 30881,
30977, 32321, 34897, 34961, 36209, 36529, 37409, 38273, 39761, 40421,
40433, 41729, 43649, 43889, 44497, 46769, 51137, 51361, 53201, 55217.
17:- 1429, 4523, 6563, 12037, 13499, 17987, 19211, 20333, 21149, 29819, 33797,
39509, 42467, 42773, 43963, 59093, 62459, 68443, 80173, 118253, 119069.
18:- 127, 1657, 3943, 7057, 8353, 8929, 11287, 11953, 12007, 15679, 17191,
20143, 23599, 25759, 26407, 28927, 29863, 31177, 32833, 33247, 33751,
35281, 38287, 39367, 39799, 41617, 46567, 47791, 51607, 55639, 59887.
19:- (4371), 6043, 15467, 21851, 22573, 78509, 82499, 95989, 96331, 100853.
20:- 3121, 3361, 3761, 8161, 9521, 21601, 39161, 41521, 44201, 51241, 55001,
69001, 69481, 70321, 73361, 74521, 78401, 86201, 90641, 92681, 97961.
21:- 29611, 33013, 47797, 49603, 52291, 61909, 65437, 70309, 95803, 98323.
22:- 1321, 6337, 7151, 13553, 16831, 18217, 18679, 29327, 34871, 38039,
39359, 40151, 43913, 54583, 60919, 65407, 66529, 68993, 72271, 79201.
23:- (645), 18539, 20747, 51797, 70381, 85469, 93979.
24:- 601, 1777, 2833, 4057, 5209, 5737, 7753, 8713, 8761, 9649, 11113,
11329, 12241, 12841, 15289, 16729, 21481, 22777, 23017, 23209, 23833,
28297, 28393, 29569, 32353, 32713, 34513, 35449, 36793, 36913, 42193,
46681. 49633, 53113, 57601, 59113, 62233, 62473, 63241, 63913, 65929.
25:- 15451, 18451, 22051, 25301, 31051, 32051, 39301, 49451, 56101, 59051.
26:- 14327, 31799, 37441, 45553, 49921.
27:- 2971, 5347, 8317, 19819.
28:- 2857, 7393, 26041, 26881, 42953.
29:- 72269.
30:- 3391, 9511, 16111, 17401, 24391, 26431, 28111, 33871, 36871, 40471,
40591, 44071, 50671, 52561, 52711, 57271, 61231, 71671, 75991, 76831.
31:- 683, 1613, 14323.
32:- 2593, 10657, 12641, 21569, 26849, 31649, 33569, 35201, 45697, 46049,
47713, 50753, 53633, 56417, 56737, 74209, 99041.
33:- 17029, 30757, 47653.
34:- (341), 2687, 9623, 9929, 12479, 34273, 42569.
35:- 42701, 82531, 86381, (91001).
36:- 11161, 31249, 48313.
37:- 13099.
38:- 1103, 7487, 12503, 46703.
39:- 71293.
40:- 13121, 15241, 18041, 23761, 29761, 31721, 35081, 37321, 38321, 40841,
43721, 50441, 57881, 58921, 60521, 66041, 66841, 70481, 74561, 81001.
41:- 17467.
42:- 2143, 6679, (10585), 20959, 21169, 39607.
43:- 83077.
44:- (2465), 25609, 31153.
45:- 5581, (8911), 21061, 25741, 26371.
46:- (1105), 5153.
47:- (2821), 26227.
48:- (1729), 2113, 4177, 15073, 23041, 27409, 53617, 57649, 59809, 65617.
49:- 51941.
50:- 2351, 27551, 27751, 38351, 46351, 51001, 107201, 115151, 121351.
51:- 67219.
52:- 26417.
53:- 91373.
54:- 18199, 47521, 54919, 71551, 75169, 88993.
55:- 48731.
56:- 6553, 15121, (25761), 28001, 30241, 35897, 40153, 41161.
57:- 15619, 33403.
58:- 3191, 16183, 18503, 34511.
59:- 838037.
60:- 27961, 37201.
61:- 103091.
62:- 10169, 46439, (62745).
63:- 103573.
64:- 6529, (12801), 15809, 25601.
65:- 75011.
66:- 52009.
67:- 13669.
68:- (1905), 114377.
69:- 161323.
70:- 8681, 38431.
71:- 171253.
72:- 1801, 2089, 15193. 46441, 47737.
73:- 472019.
74:- 11471.
75:- (2701), 119851.
76:- 93329.
77:- (1387), 107339.
78:- 49999.
79:- 17539.
80:- 49201.
81:- 4051, 8101.
82:- 18287, 42641.
83:- (83665), 292493.
84:- 169177.
85:- 29581.
86:- 141041.
87:- 304501.
88:- 59753.
89:- 167677.
90:- 23671, 27361, 42841.
91:- 46957.
92:- 49681.
93:- 244219.
94:- 312551.
95:- 115331.
96:- 4513, 37537.
97:- 1618931.
98:- 120247.
99:- 1297693.
100:-41201.
105:-2731.
The smallest odd prime p such that n = (p-1)/ord_p(2). Example:- 3 = (7-1)/2.
These are the start numbers for each sequence of multiplicative order n:-
3, 7, 43, 113, 251, 31, 1163, 73, 397, 151, 331, 1753, 4421, 631, 3061, 257,
1429, 127, 6043, 3121, 29611, 1321, 18539, 601, 15451, 14327, 2971, 2857,
72269, 3391, 683, 2593, 17029, 2687, 42701, 11161, 13099, 1103, 71293, 13121,
17467, 2143, 83077, 25609, 5581, 5153, 26227, 2113, 51941, 2351, 67219, 26417,
91373, 18199, 48731, 6553, 15619, 3191, 838037, 27961, 103091, 10169, 6529,
75011, 52009, 13669, 114377, 161323, 8681, 171253, 1801, 472019, 11471, 119851,
93329, 107339, 49999, 17539, 49201, 4051, 18287, 292493, 169177, 29581, 141041,
304501, 59753, 167677, 23671, 46957, 49681, 244219, 312551, 115331, 4513,
1618931, 120247, 1297693, 41201.
The number n such that n = (P-1)/ord_(2) for successive Poulet numbers P.
The first 100 Poulet numbers P:-
341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701,
2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321,
8481, 8911, 10261, 10585, 11305, 12801, 13741, 13747, 13981, 14491,
15709, 15841, 16705, 18705, 18721, 19951, 23001, 23377, 25761, 29341,
30121, 30889, 31417, 31609, 31621, 33153, 34945, 35333, 39865, 41041,
41665, 42799, 46657, 49141, 49981, 52633, 55245, 57421, 60701, 60787,
62745, 63973, 65077, 65281, 68101, 72885, 74665, 75361, 80581, 83333,
83665, 85489, 87249, 88357, 88561, 90751, 91001, 93961, 101101, 104653,
107185, 113201, 115921, 121465, 123251, 126217, 129889, 129921, 130561,
137149, 149281, 150851, 154101, 157641, 158369, 162193, 162401, 164737,
172081, 176149.
Corresponding values for m:-
10, 40, 28, 24, 18, 36, 28, 11, 58, 36,
60, 28, 36, 16, 230, 15, 14, 660, 36, 52,
80, 198, 30, 252, 72, 200, 60, 58, 20, 42,
22, 45, 48, 28, 96, 70, 40, 48, 460, 180,
60, 3432, 88, 72, 102, 112, 168, 44, 264, 60,
192, 21, 144, 156, 30, 153, 28, 180, 100, 22,
1012, 36, 58, 48, 60, 28, 612, 120, 60, 166,
1008, 52, 532, 148, 9840, 75, 2600, 360, 300, 76,
48, 100, 72, 168, 58, 36, 96, 70, 68, 66,
60, 50, 460, 280, 28, 48, 1160, 64, 60, 126.
Corresponding values for n:-
34, 14, 23, 46, 77, 48, 68, 186, 44, 75,
47, 117, 112, 273, 19, 312, 390, 10, 221, 160,
106, 45, 342, 42, 157, 64, 229, 237, 699, 345,
714, 352, 348, 668, 195, 285, 575, 487, 56, 163,
502, 9, 357, 439, 310, 296, 208, 803, 151, 684,
217, 2038, 324, 315, 1666, 344, 1973, 319, 607, 2763,
62, 1777, 1122, 1360, 1135, 2603, 122, 628, 1343, 502,
83, 1644, 164, 597, 9, 1210, 35, 261, 337, 1377,
2233, 1132, 1610, 723, 2125, 3506, 1353, 1856, 1920, 2078,
2488, 3017, 335, 563, 5656, 3379, 140, 2574, 2868, 1398.
For each Poulet number P listed above, let (P-1)/n = m = ord_(2),
to be consistent with the terminology used on the OEIS website.
Then 2^m is congruent to 1 modulo P. For odd m, it is shown in
parentheses. H1 indicates the HCF of one less than the factors,
suitably paired up where necessary. H2 is the HCF of H1 and m.
So H1 = hcf(f1-1,f2-1) and H2 = hcf(H1,m), where f1 and f2 are
the two suitably paired factors of P minus one. They are shown
by the parentheses in the cases where P has more than two prime
factors. Examples:-
0:- P(1) = 341 = 11.31, m = 10, n = 34, 2^10 = 1 (341) since
2^((341-1)/34 = 2^10 = 1024 = 1023+1 = (3.11.31)+1 = (3.341)+1 = 1 (341)
1:- P(18) = (7.23).41 = 6601, m = 660, n = 10, 2^660 = 1 (6601)
2:- P(26) = (3.17).251 = 12801, m = 200, n = 64, 2^200 = 1 (12801)
3:- P(37) = (3.17).(11.41) = 23001, m = 40, n = 575, 2^40 = 1 (23001)
4:- P(42) = (17.23).79 = 30889, m = 3432, n = 9, 2^3432 = 1 (30889)
5:- P(50) = (7.11.13).41 = 41041, m = 60, n = 684, 2^60 = 1 (41041)
6:- P(75) = (11.83).97 = 88561, m = 9840, n = 9, 2^9840 = 1 (88561)
7:- P(77) = (17.53).101 = 91001, m = 2600, n = 35, 2^2600 = 1 (91001)
8:- P(79) = (7.11.13).101 = 101101, m = 300, n = 337, 2^300 = 1 (101101)
9:- P(119) = 401.601 = 241001, m = 200, n = 1205, 2^200 = 1 (241001)
10:- P(152) = (31.61).211 = 399001, m = 420, n = 950, 2^420 = 1 (399001)
11:- P(153) = (7.11.13).401 = 401401, m = 600, n = 669, 2^600 = 1 (401401)
12:- P(154) = (41.73).137 = 410041, m = 3060, n = 134, 2^3060 = 1 (410041)
13:- P(178) = (11.47).1033 = 534061, m = 29670, n = 18, 2^29670 = 1 (534061)
14:- P(285) = (23.29).1999 = 1333333, m = 102564, n = 13, 2^102564 = 1 (1333333)
15:- P(717) = (13.293).2381 = 9069229, m = 104244, n = 87, 2^104244 = 1 (9069229).
# P f1.f2 m n H1 H2
01 341 11.31 10 34 10 10
02 561 (3.11).17 40 14 16 8
03 645 (3.5).43 28 23 14 14
04 1105 (5.13).17 24 46 16 8
05 1387 19.73 18 77 18 18
06 1729 (7.13).19 36 48 18 18
07 1905 (3.5).127 28 68 14 14
08 2047 23.89 (11) 186 22 11
09 2465 (5.17).29 56 44 28 28
10 2701 37.73 36 75 36 36
11 2821 (7.13).31 60 47 30 30
12 3277 29.113 28 117 28 28
13 4033 37.109 36 112 36 36
14 4369 17.257 16 273 16 16
15 4371 (3.31).47 230 19 46 46
16 4681 31.151 (15) 312 30 15
17 5461 43.127 14 390 14 14
18 6601 (7.23).41 660 10 40 20
19 7957 73.109 36 221 36 36
20 8321 53.157 52 160 52 52
21 8481 (3.11).257 80 106 32 16
22 8911 (7.19).67 198 45 66 66
23 10261 31.331 30 342 30 30
24 10585 (5.29).73 252 42 72 36
25 11305 (5.17).(7.19) 72 157 12 12
26 12801 (3.17).251 200 64 50 50
27 13741 (7.13).151 60 229 30 30
28 13747 59.233 58 237 58 58
29 13981 (11.31).41 20 699 20 20
30 14491 43.337 42 345 42 42
31 15709 23.683 22 714 22 22
32 15841 (7.13).73 (45) 352 18 9
33 16705 (5.13).257 48 348 64 16
34 18705 (3.5).(29.43) 28 668 14 14
35 18721 97.193 96 195 96 96
36 19951 71.281 70 285 70 70
37 23001 (3.17).(11.41) 40 575 50 10
38 23377 97.241 48 487 48 48
39 25761 (3.31).277 460 56 92 92
40 29341 (13.37).61 180 163 60 60
41 30121 (7.13).331 60 502 30 30
42 30889 (17.23).79 3432 9 78 78
43 31417 89.353 88 357 88 88
44 31609 73.433 72 439 72 72
45 31621 103.307 102 310 102 102
46 33153 (3.43).257 112 296 128 16
47 34945 (5.29).241 168 208 48 24
48 35333 89.397 44 397 44 44
49 39865 (5.7.17).67 264 151 66 66
50 41041 (7.11.13).41 60 684 40 20
51 41665 (5.13).641 192 217 64 64
52 42799 127.337 (21) 2038 14 7
53 46657 (13.37).97 144 324 96 48
54 49141 157.313 156 315 156 156
55 49981 151.331 30 1666 30 30
56 52633 (7.73).103 (153) 344 34 17
57 55245 (3.5).(29.127) 28 1973 14 14
58 57421 (7.13).631 180 319 90 90
59 60701 101.601 100 607 100 100
60 60787 89.683 22 2793 22 22
61 62745 (3.5.47).89 1012 62 88 44
62 63973 (7.13).(19.37) 36 1777 18 18
63 65077 59.1103 58 1122 58 58
64 65281 97.673 48 1360 96 48
65 68101 (11.41).151 60 1135 150 30
66 72885 (3.5.43).113 28 2603 28 28
67 74665 (5.109).137 612 122 136 68
68 75361 (11.13.17).31 120 628 30 30
69 80581 61.1321 60 1343 60 60
70 83333 167.499 166 502 166 166
71 83665 (5.29).577 1008 83 144 144
72 85489 53.1613 52 1644 52 52
73 87249 (3.127).229 532 164 76 76
74 88357 149.593 148 597 148 148
75 88561 (11.83).97 9840 9 48 48
76 90751 151.601 (75) 1210 150 75
77 91001 (17.53).101 2600 35 100 100
78 93961 (7.31).433 360 261 216 72
79 101101 (7.11.13).101 300 337 100 100
80 104653 229.457 76 1377 228 76
81 107185 (5.13.17).97 48 2233 48 48
82 113201 (11.41).251 100 1132 50 50
83 115921 (13.37).241 72 1610 240 24
84 121465 (5.17).1429 168 723 84 84
85 123251 59.2089 58 2125 58 58
86 126217 (7.13.19).73 36 3506 72 36
87 129889 193.673 96 1353 96 96
88 129921 (3.127).(11.31) 70 1856 20 10
89 130561 137.953 68 1920 136 68
90 137149 (23.67).89 66 2078 44 22
91 149281 (11.41).331 60 2488 30 30
92 150851 251.601 50 3017 50 50
93 154101 (3.31).1657 460 335 92 92
94 157641 (3.11.17).281 280 563 280 280
95 158369 29.(43.127) 28 5656 28 28
96 162193 241.673 48 3379 48 48
97 162401 (17.41).233 1160 140 232 232
98 164737 257.641 64 2574 128 64
99 172081 (7.13).(31.61) 60 2868 90 30
100 176149 (19.73).127 126 1398 126 126
101 181901 101.1801 100 1819 100 100
102 188057 89.2113 44 4274 88 44
103 188461 (7.13.19).109 36 5235 108 36
104 194221 167.1163 166 1170 166 166
105 196021 (7.41).683 660 297 22 22
106 196093 157.1249 156 1257 156 156
107 204001 (7.151).193 480 425 96 96
108 206601 (3.17).4051 200 1033 50 50
109 208465 (5.241).173 1032 202 172 172
110 212421 (3.11.157).41 260 817 20 20
111 215265 (3.5).(113.127) 28 7688 14 14
112 215749 79.2731 78 2766 78 78
113 219781 271.811 270 814 270 270
114 220729 103.2143 (51) 4328 102 51
115 223345 (5.19).2351 1692 132 94 94
116 226801 337.673 336 675 336 336
117 228241 (13.97).181 720 317 180 180
118 233017 43.5419 42 5548 42 42
119 241001 401.601 200 1205 200 200
120 249841 433.577 144 1735 16 16
121 252601 (41.61).101 300 842 100 100
122 253241 157.1613 52 4870 52 52
123 256999 233.1103 (29) 8862 58 29
124 258511 (11.331).71 210 1231 70 70
125 264773 149.1777 148 1789 148 148
126 266305 (5.13).(17.241) 24 11096 64 8
127 271951 31.8761 (75) 3626 30 15
128 272251 (7.19.23).89 198 1375 22 22
129 275887 263.1049 262 1053 262 262
130 276013 (19.73).199 198 1394 198 198
131 278545 (5.17).(29.113) 56 4974 84 28
132 280601 277.1013 92 3050 92 92
133 282133 307.919 306 922 306 306
134 284581 (11.41).631 180 1581 30 30
135 285541 (31.151).61 60 4759 60 60
136 289941 (3.127).761 2660 109 380 380
137 294409 (37.73).109 36 8178 108 36
139 314821 (13.61).397 660 477 396 132
140 318361 241.1321 120 2653 120 120
141 323713 (13.37).673 144 2248 32 16
142 332949 (3.29.43).89 308 1081 44 44
143 334153 (19.43).409 4284 78 408 204
144 340561 (13.17).(23.67) 264 1290 44 44
145 341497 (13.109).241 72 4743 24 24
146 348161 (11.31).1021 340 1024 340 340
147 357761 131.2731 130 2752 130 130
148 367081 (11.13.17).151 120 3059 30 30
149 387731 (43.127).71 70 5539 70 70
150 390937 313.1249 156 2506 312 156
151 396271 223.1777 74 5355 222 74
152 399001 (31.61).211 420 950 210 210
153 401401 (7.11.13).401 600 669 200 200
154 410041 (41.73).137 3060 134 136 68
155 422659 163.2593 162 2609 162 162
156 423793 (17.257).97 48 8829 48 48
157 427233 (3.53).2687 4108 104 158 158
158 435671 191.2281 190 2293 190 190
159 443719 167.2657 166 2673 166 166
160 448921 (11.37).1103 5220 86 58 58
161 449065 (5.19.29).163 2268 198 162 162
162 451905 (3.5.47).641 1472 307 64 64
163 452051 251.1801 50 9041 50 50
164 458989 277.1657 92 4989 276 92
165 464185 (5.17.43).127 56 8289 14 14
166 476971 (11.131).331 390 1223 30 30
167 481573 337.1429 84 5733 28 28
168 486737 233.2089 (29) 16784 232 29
169 488881 (37.73).181 180 2716 180 180
170 489997 157.3121 156 3141 156 156
171 493697 (17.257).113 112 4408 112 112
172 493885 (5.7.137).103 204 2421 102 102
173 512461 (31.61).271 540 949 270 270
174 513629 293.1753 292 1759 292 292
175 514447 359.1433 (179) 2874 358 179
176 526593 (3.257).683 176 2992 22 22
177 530881 (13.97).421 1680 316 420 420
178 534061 (11.47).1033 29670 18 516 258
179 552721 (13.17.41).61 120 4606 60 60
180 556169 457.1217 152 3659 152 152
181 563473 (37.97).157 1872 301 156 156
182 574561 (13.193).229 1824 315 228 228
183 574861 (7.41).2003 8580 67 286 286
184 580337 499.1163 166 3496 166 166
185 582289 113.5153 112 5199 112 112
186 587861 443.1327 442 1330 442 442
187 588745 (5.1613).73 468 1258 36 36
188 604117 389.1553 388 1557 388 388
189 611701 151.4051 150 4078 150 150
190 617093 (43.127).113 28 22039 28 28
191 622909 (7.23.73).53 5148 121 52 52
192 625921 (31.61).331 60 10432 30 30
193 635401 (13.37).1321 180 3530 120 60
194 642001 401.1601 400 1605 400 400
195 647089 79.8191 (39) 16592 78 39
196 653333 467.1399 466 1402 466 466
197 656601 (3.11.101).197 4900 134 196 196
198 657901 307.2143 102 6450 306 102
199 658801 (11.13.17).271 1080 610 270 270
200 665281 577.1153 288 2310 576 288
201 665333 283.2351 94 7078 94 94
202 665401 (11.251).241 600 1109 120 60
203 670033 (7.13.37).199 396 1692 198 198
204 672487 103.6529 102 6593 102 102
205 679729 337.2017 336 2023 336 336
206 680627 107.6361 106 6421 106 106
207 683761 (13.149).353 9768 70 176 88
208 688213 127.5419 42 16386 126 42
209 710533 487.1459 486 1462 486 486
210 711361 (7.673).151 240 2964 30 30
211 721801 601.1201 300 2406 600 300
212 722201 401.1801 200 3611 200 200
213 722261 491.1471 490 1474 490 490
214 729061 349.2089 348 2095 348 348
215 738541 67.(73.151) 990 746 66 66
216 741751 431.1721 (215) 3450 430 215
217 742813 223.3331 222 3346 222 222
218 743665 (5.13.17).673 48 15493 48 48
219 745889 353.2113 88 8476 352 88
220 748657 (7.13.19).433 72 10398 432 72
221 757945 (5.17.37).241 72 10527 24 24
222 769567 439.1753 146 5271 438 146
223 769757 227.3391 226 3406 226 226
224 786961 (7.19.61).97 720 1093 48 48
225 800605 (5.13.109).113 252 3177 28 28
226 818201 101.8101 100 8182 100 100
227 825265 (5.7.17.19).73 72 11462 72 72
228 831405 (3.5.43).1289 644 1291 644 644
229 838201 (7.13.61).151 60 13970 150 30
230 838861 397.2113 44 19065 396 44
231 841681 (19.31).1429 1260 668 28 28
232 847261 31.151.181 180 4707 180 180
233 852481 (7.631).193 1440 592 192 96
234 852841 (11.31.41).61 60 14214 60 60
235 873181 661.1321 660 1323 660 660
236 875161 (7.31.37).109 180 4862 36 36
237 877099 307.2857 102 8599 102 102
238 898705 (5.17.97).109 144 6241 36 36
239 915981 (3.11.41).677 3380 271 676 676
240 916327 479.1913 (239) 3834 478 239
241 934021 (11.19.41).109 180 5189 36 36
242 950797 (23.67).617 462 2058 154 154
243 976873 313.3121 156 6262 312 156
244 983401 331.2971 330 2980 330 330
245 997633 (7.13.19).577 144 6928 576 144
246 1004653 (13.109).709 2124 473 708 708
247 1016801 251.4051 50 20336 50 50
248 1018921 (71.127).113 140 7278 56 28
249 1023121 (11.331).281 210 4872 280 70
250 1024651 (19.199).271 2970 345 270 270
251 1033669 (7.13.37).307 612 1689 306 306
252 1050985 (5.13.19.37).23 396 2654 22 22
253 1052503 (23.683).67 66 15947 66 66
254 1052929 (17.257).241 48 21936 48 48
255 1053761 593.1777 148 7120 592 148
256 1064053 199.5347 198 5374 198 198
257 1073021 131.8191 130 8254 130 130
258 1082401 601.1801 (25) 43296 600 25
259 1082809 (7.13.73).163 324 3342 162 162
260 1092547 523.2089 522 2093 522 522
261 1093417 (13.241).349 696 1571 348 348
262 1104349 (29.113).337 84 13147 84 84
263 1106785 (5.17.29).449 224 4941 224 224
264 1109461 83.13367 82 13530 82 82
265 1128121 (31.151).241 120 9401 120 120
266 1132657 337.3361 168 6742 336 168
267 1139281 (11.13.257).31 240 4747 30 30
268 1141141 (31.281).131 910 1254 130 130
269 1145257 103.11119 (51) 22456 102 51
270 1152271 (43.127).211 210 5487 210 210
271 1157689 (13.19.109).43 252 4594 42 42
272 1168513 409.2857 204 5728 408 204
273 1193221 (31.631).61 180 6629 60 60
274 1194649 sq 1093.1093 364 3282 1092 364
275 1207361 449.2689 224 5390 448 224
276 1246785 (3.5.43).1933 644 1936 644 644
277 1251949 409.3061 204 6137 204 204
278 1252697 733.1709 244 5134 244 244
279 1275681 (3.11.31).(29.43) 140 9112 14 14
280 1277179 283.4513 94 13587 282 94
281 1293337 569.2273 568 2277 568 568
282 1302451 571.2281 570 2285 570 570
283 1306801 (19.631).109 180 7260 108 36
284 1325843 499.2657 166 7987 166 166
285 1333333 (23.29).1999 102564 13 666 666
286 1357441 673.2017 336 4040 672 336
287 1357621 (23.67).881 330 4114 220 110
288 1373653 829.1657 828 1659 828 828
289 1394185 (5.13.241).89 264 5281 88 88
290 1397419 67.20857 66 21173 66 66
291 1398101 (23.89).683 22 63550 682 22
292 1419607 (7.1459).139 11178 127 138 138
293 1433407 599.2393 598 2397 598 598
294 1441091 347.4153 346 4165 346 346
295 1457773 349.4177 348 4189 348 348
296 1459927 (7.73).2857 306 4771 34 34
297 1461241 (37.73).541 540 2706 540 540
298 1463749 (7.67).3121 1716 8533 52 52
299 1472065 (5.7.137).307 204 7216 102 102
300 1472353 (17.257).337 336 4382 336 336
301 1472505 (3.5.1103).89 1276 1154 88 44
302 1485177 (3.397).(29.43) 308 4822 14 14
303 1489665 (3.5.2113).47 1012 1472 46 46
304 1493857 547.2731 546 2736 546 546
305 1500661 61.(73.337) 1260 1191 60 60
306 1507561 (11.31).4421 340 4434 340 340
307 1507963 971.1553 194 7773 194 194
308 1509709 389.3881 388 3891 388 388
309 1520905 (5.17.29).617 616 2469 616 616
310 1529185 (5.7).43691 204 7496 34 34
311 1530787 619.2473 618 2477 618 618
312 1533601 (31.811).61 540 2840 60 60
313 1533961 (11.13.17).631 360 4261 90 90
314 1534541 (43.127).281 70 21922 140 70
315 1537381 877.1753 876 1755 876 876
316 1549411 (31.151).331 30 51647 30 30
317 1569457 (17.19.43).113 504 3114 112 56
318 1579249 (7.17.23).577 1584 997 48 48
319 1584133 727.2179 726 2182 726 726
320 1608465 (3.5.683).157 572 2812 52 52
321 1615681 (23.353).199 792 2040 198 198
322 1620385 (5.13.257).97 48 33758 96 48
323 1643665 (5.3389).97 5808 283 48 48
324 1678541 1013.1657 92 18245 92 92
325 1690501 751.2251 750 2254 750 750
326 1711381 (7.41.89).67 660 2593 66 66
327 1719601 (13.17.31).251 600 2866 50 50
328 1730977 439.3943 (219) 7904 438 219
329 1735841 761.2281 380 4568 760 380
330 1746289 37.(109.433) 72 24254 36 36
331 1755001 937.1873 936 1875 936 936
332 1773289 (7.19.67).199 198 8956 198 198
333 1801969 (13.97).1429 336 5363 84 84
334 1809697 673.2689 672 2693 672 672
335 1811573 389.4657 388 4669 388 388
336 1815465 (3.5.127).953 476 3814 952 476
337 1826203 337.5419 42 43481 42 42
338 1827001 (11.41).4051 100 18270 450 50
339 1830985 (5.13.17).1657 552 3317 552 552
340 1837381 (7.13.331).61 60 30623 60 60
341 1839817 (7.607).433 7272 253 72 72
342 1840357 (43.337).127 42 43818 126 42
343 1857241 (31.331).181 180 10318 180 180
344 1876393 613.3061 612 3066 612 612
345 1892185 (5.17.113).197 392 4827 196 196
346 1896961 (11.331).521 780 2432 520 260
347 1907851 (11.251).691 1150 1659 690 230
348 1908985 (5.13.683).43 924 2066 42 42
349 1909001 (41.101).461 2300 830 460 460
350 1937881 (11.17.241).43 840 2307 42 42
351 1969417 919.2143 (153) 12872 306 153
352 1987021 997.1993 996 1995 996 996
353 1993537 (7.13.19).1153 288 6922 576 288
354 1994689 577.3457 576 3463 576 576
355 2004403 307.6529 102 19651 102 102
356 2008597 709.2833 708 2837 708 708
357 2035153 1009.2017 1008 2019 1008 1008
358 2077545 (3.5.43).3221 644 3226 644 644
359 2081713 373.5581 372 5596 372 372
360 2085301 (41.181).281 1260 1655 140 140
361 2089297 (7.19.23).683 198 10552 22 22
362 2100901 (11.31.61).101 300 7003 100 100
363 2113665 (3.5.29.43).113 28 75488 112 28
364 2113921 (19.31.37).97 720 2936 96 48
365 2121301 (7.13).23311 180 11785 90 90
366 2134277 1193.1789 596 3581 596 596
367 2142141 (3.71.113).89 1540 1391 44 44
368 2144521 (29.73).1013 5796 370 92 92
369 2162721 (3.11).65537 160 13517 32 32
370 2163001 1201.1801 300 7210 600 300
371 2165801 (11.401).491 9800 221 490 490
372 2171401 (41.251).211 2100 1034 210 210
373 2181961 661.3301 660 3306 660 660
374 2184571 523.4177 522 4185 522 522
375 2205967 743.2969 (371) 5946 742 371
376 2213121 (3.31.449).53 14560 152 52 52
377 2232865 (5.17.109).241 72 31012 48 24
378 2233441 (7.151).2113 660 3384 1056 132
379 2261953 673.3361 336 6732 336 336
380 2264369 389.5821 388 5836 388 388
381 2269093 953.2381 476 4767 476 476
382 2284453 1069.2137 1068 2139 1068 1068
383 2288661 (3.23.41).809 22220 103 404 404
384 2290641 (3.257).2971 880 2603 22 22
385 2299081 (43.127).421 420 5474 420 420
386 2304167 1103.2089 (29) 79454 58 29
387 2313697 449.5153 224 10329 224 224
388 2327041 (37.109).577 144 16160 576 144
389 2350141 (31.1613).47 5980 393 46 46
390 2387797 773.3089 772 3093 772 772
391 2414001 (3.31.257).101 400 6035 100 100
392 2419385 (5.229).2113 836 2894 88 44
393 2433601 (17.37.73).53 936 2600 52 52
394 2434651 601.4051 50 48693 150 50
395 2455921 (13.19.61).163 1620 1516 162 162
396 2487941 911.2731 182 13670 910 182
397 2491637 1289.1933 644 3869 644 644
398 2503501 (7.11.13.41).61 60 41725 60 60
399 2508013 (53.79).599 3588 699 299 299
400 2510569 709.3541 708 3546 708 708
401 2513841 (3.11.17).4481 560 4489 560 560
402 2528921 41.61681 40 63223 40 40
403 2531845 (5.19.29).919 4284 591 34 34
404 2537641 (17.1321).113 840 3021 56 56
405 2603381 (11.761).311 11780 221 310 310
406 2609581 (13.4271).47 84180 31 46 46
407 2615977 (7.13.17.19).89 792 3303 88 88
408 2617451 881.2971 110 23795 110 110
409 2626177 (17.241).641 192 13678 128 64
410 2628073 (7.37.73).139 828 3174 46 46
411 2649029 997.2657 332 7979 332 332
412 2649361 (11.13.97).191 4560 581 38 38
413 2670361 409.6529 204 13090 408 204
414 2704801 (11.29.61).139 9660 280 46 46
415 2719981 (11.37.41).163 1620 1679 162 162
416 2722681 (13.19.73).151 180 15126 30 30
417 2746477 829.3313 828 3317 828 828
418 2746589 677.4057 676 4063 676 676
419 2748023 479.5737 (239) 11498 478 239
420 2757241 461.5981 460 5994 460 460
421 2773981 (7.19).20857 198 14010 132 22
422 2780731 (31.331).271 270 10299 270 270
423 2793351 (3.11.1801).47 1150 2429 46 46
424 2797921 (7.19.109).193 288 9715 96 96
425 2811271 881.3191 (55) 51114 110 55
426 2827801 73.38737 72 39275 72 72
427 2867221 (7.31.73).181 180 15929 180 180
428 2880361 (11.17.73).211 2520 1143 210 210
429 2882265 (3.5.17.127).89 616 4679 88 88
430 2899801 (37.433).181 360 8055 180 180
431 2909197 293.9929 292 9963 292 292
432 2921161 (17.23.31).241 1320 2213 120 120
433 2940337 (17.257).673 48 61257 336 48
434 2944261 991.2971 990 2974 990 990
435 2953711 (31.151).631 (45) 65638 90 45
436 2976487 863.3449 (431) 6906 862 431
437 2977217 1409.2113 704 4229 704 704
438 2987167 (43.127).547 546 5471 546 546
439 3020361 (3.31.47).691 230 13132 230 230
440 3048841 61.(151.331) 60 50814 30 30
441 3057601 (43.337).211 210 14560 210 210
442 3059101 1237.2473 1236 2475 1236 1236
443 3073357 (7.67).6553 2574 1194 52 52
444 3090091 1163.2657 166 18615 166 166
445 3094273 (7.13.37).919 612 5056 306 306
446 3116107 883.3529 882 3533 882 882
447 3125281 1021.3061 1020 3064 1020 1020
448 3146221 (13.31.37).211 1260 2497 210 210
449 3165961 (17.43.61).71 840 3769 70 70
450 3181465 (5.7.17).5347 792 4017 594 198
451 3186821 (11.281).1031 7210 442 1030 1030
452 3224065 (5.13.257).193 96 33584 192 96
453 3225601 (71.251).181 6300 512 180 180
454 3235699 1609.2011 402 8049 402 402
455 3316951 (11.331).911 2730 1215 910 910
456 3336319 (7.73).6529 306 10903 102 102
457 3337849 409.8161 408 8181 408 408
458 3345773 1181.2833 236 14177 236 236
459 3363121 1297.2593 648 6190 1296 648
460 3370641 (3.17.29.43).53 728 4630 52 52
461 3375041 1061.3181 1060 3184 1060 1060
462 3375487 919.3673 918 3677 918 918
463 3400013 1597.2129 532 6391 532 532
464 3413533 307.11119 102 33466 102 102
465 3429037 157.21841 156 21981 156 156
466 3435565 (5.7.953).103 204 16841 102 102
467 3471071 311.11161 310 11197 310 310
468 3539101 941.3761 940 3765 940 940
469 3542533 1087.3259 1086 3262 1086 1086
470 3567481 311.11471 155 23016 310 155
471 3568661 1091.3271 1090 3274 1090 1090
472 3581761 (29.113).1093 364 9840 1092 364
473 3605429 137.26317 68 53021 68 68
474 3656449 577.6337 288 12696 576 288
475 3664585 (5.29.127).199 2772 1322 198 198
476 3679201 (29.433).293 36792 100 292 292
477 3726541 (7.13).(31.1321) 60 62109 90 30
478 3746289 (3.43.257).113 112 33449 112 112
479 3755521 (7.11.17.19).151 360 10432 30 30
480 3763801 (83.137).331 41820 90 30 30
481 3779185 (5.17.257).173 688 5493 172 172
482 3814357 53.(79.911) 1092 3493 52 52
483 3828001 (101.251).151 300 12760 150 150
484 3898129 1249.3121 156 24988 624 156
485 3911197 223.17539 222 17618 222 222
486 3916261 (19.109).(31.61) 180 21757 90 90
487 3936691 (11.2143).167 42330 93 166 166
488 3985921 1153.3457 576 6920 1152 576
489 4005001 (7.11.13).4001 3000 1335 1000 1000
490 4014361 (13.2833).109 2124 1890 108 108
491 4025905 (5.13.257).241 48 83873 48 48
492 4038673 (17.673).353 528 7649 176 176
493 4069297 (37.109).1009 504 8074 1008 504
494 4072729 67.(89.683) 66 61708 66 66
495 4082653 1429.2857 1428 2859 1428 1428
496 4097791 271.15121 270 15177 270 270
497 4101637 1013.4049 1012 4053 1012 1012
498 4151869 1289.3221 644 6447 644 644
499 4154161 (11.41.151).61 60 69236 60 60
500 4154977 (19.379).577 3024 1374 288 144
501 4181921 1181.3541 236 17720 1180 236
502 4188889 431.9719 (43) 97416 86 43
503 4209661 (71.281).211 210 20046 210 210
504 4229601 (3.41.137).251 1700 2488 50 50
505 4259905 (5.13).65537 96 44374 64 32
506 4314967 1039.4153 1038 4157 1038 1038
507 4335241 (53.157).521 260 16674 520 260
508 4360621 1321.3301 660 6607 660 660
509 4361389 853.5113 852 5119 852 852
510 4363261 (7.23.41).661 660 6611 660 660
511 4371445 (5.13.109).617 2772 1577 44 44
512 4415251 1051.4201 1050 4205 1050 1050
513 4463641 (7.13.181).271 540 8266 270 270
514 4469471 1831.2441 (305) 14654 610 305
515 4480477 547.8191 546 8206 546 546
516 4502485 (5.13.113).613 4284 1051 612 612
517 4504501 (19.79).3001 58500 77 1500 1500
518 4513841 1021.4421 340 13276 340 340
519 4535805 (3.5.127).2381 476 9529 476 476
520 4567837 1069.4273 1068 4277 1068 1068
521 4613665 (5.7.193).683 1056 4369 22 22
522 4650049 (37.109).1153 288 16146 576 288
523 4670029 (7.13.19).(37.73) 36 129723 108 36
524 4682833 (29.113).1429 84 55748 84 84
525 4698001 (7.11.17.37).97 720 6525 48 48
526 4706821 (7.61.73).151 180 26149 60 60
527 4714201 (3.317).751 360 13095 50 10
528 4767841 (13.19.97).199 1584 3010 198 198
529 4806061 (13.41.71).127 420 11443 42 42
530 4827613 (7.593).1163 36852 131 166 166
531 4835209 239.20231 119 40632 238 119
532 4863127 1103.4409 (551) 8826 1102 551
533 4864501 (37.73).1801 900 5405 900 900
534 4868701 601.8101 100 48687 300 100
535 4869313 337.14449 84 57968 336 84
536 4877641 (73.109).613 612 7970 612 612
537 4895065 (5.7.17.19).433 72 67987 72 72
538 4903921 (11.31.73).197 8820 556 196 196
539 4909177 (7.13.73).739 1476 3326 738 738
540 4917331 (23.67).3191 330 14901 110 110
541 4917781 (11.37.43).281 1260 3903 140 140
542 4922413 839.5867 838 5874 838 838
543 4974971 (43.127).911 182 27335 182 182
544 4984001 (11.41.257).43 560 8900 14 14
545 5016191 647.7753 (323) 15530 646 323
546 5031181 (19.23.29).397 2772 1815 396 396
547 5034601 (11.13.17).(19.109) 360 13985 90 90
548 5044033 1297.3889 648 7784 1296 648
549 5049001 (31.271).601 (675) 7480 600 75
550 5095177 1129.4513 564 9034 1128 564
551 5131589 617.8317 308 16661 308 308
552 5133201 (3.17.251).401 200 25666 400 200
553 5148001 (41.241).521 1560 3300 520 520
554 5173169 1093.4733 364 14212 364 364
555 5173601 929.5569 464 11150 928 464
556 5176153 1609.3217 804 6438 1608 804
557 5187637 (7.13.109).523 1044 4969 522 1044
558 5193721 (13.17.71).331 840 6183 30 30
559 5250421 (11.109.151).29 1260 4167 28 14
560 5256091 811.6481 810 6489 810 810
561 5258701 (7.41.73).251 900 5843 50 50
562 5271841 (73.257).281 5040 1046 280 280
563 5284333 227.23279 226 23382 226 226
564 5310721 (13.37.61).181 180 29504 180 180
565 5351537 1889.2833 472 11338 944 472
566 5400489 (3.283).6361 4982 1084 424 106
567 5423713 2017.2689 672 8071 672 672
568 5444489 (29.197).953 3332 1634 952 476
569 5456881 (17.257).1249 624 8745 624 624
570 5481451 (31.151).1171 1170 4685 1170 1170
571 5489121 (3.11.41).4057 3380 1624 1352 676
572 5489641 1657.3313 828 6630 1656 828
573 5524693 1777.3109 444 12443 444 444
574 5529745 (5.13.241).353 264 20946 176 88
575 5545145 (5.17.89).733 5368 1033 244 244
576 5551201 (31.331).541 540 10280 540 540
577 5560809 (3.59.89).353 2552 2179 88 88
578 5575501 1181.4721 1180 4725 1180 1180
579 5590621 409.13669 204 27405 204 204
580 5599765 (5.37).30269 828 6763 92 92
581 5632705 (5.13.449).193 672 8382 192 96
582 5672041 661.8581 660 8594 660 660
583 5681809 (7.73).11119 (153) 37136 102 51
584 5733649 (19.577).523 4176 1373 522 522
585 5758273 (19.37).8191 468 12304 26 26
586 5766001 1201.4801 1200 4805 1200 1200
587 5804821 (11.41.61).211 420 13821 210 210
588 5859031 (31.331).571 570 10279 570 570
589 5872361 (11.17.31).1013 920 6383 92 92
590 5919187 1777.3331 222 26663 222 222
591 5968261 (13.19.331).73 180 33157 36 36
592 5968873 (43.127).1093 364 16398 1092 364
593 5977153 (7.13.19).3457 576 10377 1728 576
594 6027193 1553.3881 388 15534 776 388
595 6049681 (11.31.113).157 1820 3324 52 52
596 6054985 (5.53.73).313 468 12938 312 156
597 6118141 71.86171 70 87402 70 70
598 6122551 2143.2857 102 60025 714 102
599 6135585 (3.5.449).911 2912 2107 182 182
600 6140161 937.6553 (117) 52480 936 117
601 6159301 211.29191 210 29330 210 210
602 6183601 (31.1321).151 60 103060 150 30
603 6189121 (61.241).421 840 7368 420 420
604 6212361 (3.17.41).2971 440 14119 110 110
605 6226193 1933.3221 644 9668 644 644
606 6233977 1117.5581 1116 5586 1116 1116
607 6235345 (5.17.673).109 144 43301 36 36
608 6236257 1249.4993 624 9994 1248 624
609 6236473 (23.683).397 44 141738 44 44
610 6242685 (3.5.29.127).113 28 222953 28 28
611 6255341 (113.281).197 980 6383 196 196
612 6278533 1447.4339 1446 4342 1446 1446
613 6309901 (13.73.109).61 180 35055 60 60
614 6313681 (11.17.19).1777 13320 474 1776 888
615 6334351 727.8713 363 17450 726 363
616 6350941 (41.191).811 10260 619 90 90
617 6368689 1129.5641 564 11292 1128 564
618 6386993 653.9781 652 9796 652 652
619 6474691 (31.331).631 90 71941 90 90
620 6539527 1279.5113 1278 5117 1278 1278
621 6617929 (47.1013).139 276 23978 138 138
622 6628385 (5.17.29).2689 224 29591 224 224
623 6631549 463.14323 462 14354 462 462
624 6658669 1999.3331 666 9998 666 666
625 6732817 (7.13.241).307 408 16502 102 102
626 6733693 (109.163).379 2268 2969 378 378
627 6749021 1061.6361 1060 6367 1060 1060
628 6779137 2017.3361 336 20176 672 336
629 6787327 1303.5209 651 10426 1302 651
630 6836233 313.21841 156 43822 312 156
631 6840001 (7.17.229).251 11400 600 250 50
632 6868261 (43.211).757 3780 1817 756 756
633 6886321 (13.97).(43.127) 336 20495 420 84
634 6912079 1753.3943 438 15781 146 146
639 6998881 113.(241.257) 336 20830 112 112
640 7008001 (7.11.13).7001 1500 4672 1000 500
641 7017193 937.7489 468 14994 936 468
642 7040001 (3.257).(23.397) 176 40000 110 22
643 7177105 5.13.109.1013 828 8668 1012 92
644 7207201 (17.353).1201 660 1092 1200 60
645 7215481 (7.13.37).2143 612 11790 306 306
646 7232321 1553.4657 388 18640 1552 388
647 7233265 (5.13.257).433 144 50231 144 144
648 7259161 (7.13.331).241 120 60493 120 120
649 7273267 (23.683).463 462 15743 462 462
650 7295851 1801.4051 50 145917 150 50
651 7306261 2341.3121 780 9367 780 780
652 7306561 1153.6337 288 25370 576 288
653 7414333 229.32377 228 32519 228 228
654 7416289 (19.73).5347 198 37456 198 198
655 7428421 (7.11.13.41).181 180 41269 180 180
656 7429117 107.69431 106 70086 106 106
657 7455709 (73.109).937 468 15931 468 468
658 7462001 911.8191 (91) 82000 910 91
659 7516153 (19.73).5419 126 59652 126 126
660 7519441 (41.241).761 2280 3298 760 760
661 7546981 (13.31.61).307 1020 7399 102 102
662 7656721 (31.673).367 14640 523 366 366
663 7674967 937.8191 117 65598 234 117
664 7693401 (3.17.251).601 200 38467 200 200
665 7724305 (5.37.43).971 24444 316 194 194
666 7725901 1051.7351 1050 7358 1050 1050
667 7759937 929.8353 464 16724 928 464
668 7803769 (43.127).1429 84 92902 84 84
669 7808593 (13.17.397).89 264 29578 88 88
670 7814401 (7.151).7393 1320 5920 352 88
671 7820201 1831.4271 305 25640 610 305
672 7883731 811.9721 810 9733 810 810
673 7995169 (7.13.103).853 14484 552 852 852
674 8012845 (5.29.73).757 756 10599 756 756
675 8036033 1637.4909 1636 4912 1636 1636
676 8041345 (5.13.641).193 192 41882 192 192
677 8043841 (13.433).1429 504 15960 28 28
678 8095447 1423.5689 711 11386 1422 711
679 8134561 (37.109).2017 1008 8070 2016 1008
680 8137585 (5.1613).1009 6552 1242 112 56
681 8137633 (7.3331).349 12876 632 348 348
682 8180461 541.15121 540 15149 540 540
683 8209657 (17.337).1433 30072 273 1432 1432
684 8231653 2029.4057 2028 4059 2028 2028
685 8239477 659.12503 658 12522 658 658
686 8280229 397.20857 132 62729 132 132
687 8321671 (31.331).811 270 30821 270 270
688 8322945 (3.5.17.257).127 112 74312 14 14
689 8341201 (11.31.61).401 600 13902 400 200
690 8355841 (13.41.61).257 240 34816 256 16
691 8362201 617.13553 616 13575 616 616
692 8384513 277.30269 92 91136 92 92
693 8388607 47.178481 23 364722 46 23
694 8462233 (13.37.241).73 72 117531 72 72
695 8534233 709.12037 708 12054 708 708
696 8640661 (31.1321).211 420 20573 210 210
697 8646121 (11.19.41).1009 2520 3431 504 504
698 8650951 1471.5881 1470 5885 1470 1470
699 8656705 (5.37.641).73 576 15029 72 72
700 8719309 (19.37.79).157 468 18631 156 156
701 8719921 (7.23.41).1321 660 13212 1320 660
702 8725753 2089.4177 87 100296 2088 87
703 8727391 71.122921 35 249354 70 35
704 8745277 2857.3061 204 42869 204 204
705 8812273 (17.257).2017 336 26227 336 336
706 8830801 (7.19.67).991 990 8920 990 990
707 8902741 1723.5167 1722 5170 1722 1722
708 8916251 (31.4051).71 350 25475 70 70
709 8927101 (31.37.43).181 1260 7085 180 180
710 8992201 (17.31.151).113 840 10705 56 56
711 9006401 1733.5197 1732 5200 1732 1732
712 9037729 2689.3361 672 13449 672 672
713 9040013 1553.5821 388 23299 388 388
714 9046297 (13.43).16183 7812 1158 558 558
715 9056501 1229.7369 1228 7375 1228 1228
716 9063105 (3.5.257).2351 752 12052 94 94
717 9069229 (13.293).2381 104244 87 476 476
718 9073513 953.9521 476 19062 952 476
719 9084223 (19.43).11119 2142 4241 34 34
720 9106141 (11.41.331).61 60 151769 60 60
721 9131401 2137.4273 1068 8550 2136 1068
722 9143821 (43.337).631 630 14514 630 630
723 9223401 (3.11.17.41).401 200 46117 200 200
724 9224391 (3.11.31.127).71 70 131777 70 70
725 9273547 1523.6089 1522 6093 1522 1522
726 9345541 59.(151.1049) 113970 82 58 58
727 9371251 1531.6121 1530 6125 1530 1530
728 9439201 (61.271).571 10260 920 570 570
729 9480461 2971.3191 110 6186 110 110
730 9494101 (23.67).(61.101) 3300 2877 1540 220
731 9533701 1783.5347 1782 5350 1782 1782
732 9564169 619.15451 618 15476 618 618
733 9567673 509.18797 508 18834 508 508
734 9582145 (5.23.97).859 6864 1396 858 858
735 9585541 (7.31.163).271 810 11834 270 270
736 9588151 79.121369 39 245850 78 39
737 9591661 1171.8191 1170 8198 1170 1170
738 9613297 (19.29.73).239 4284 2244 238 238
739 9638785 (5.13.257).577 144 66936 576 144
740 9692453 (23.683).616 154 62938 308 154
741 9724177 673.14449 336 28941 336 336
742 9729301 1201.8101 300 32431 300 300
743 9774181 181.54001 180 54301 180 180
744 9816465 (3.5.127).5153 112 87647 112 112
745 9834781 (11.31.151).191 570 17254 190 190
746 9863461 2221.4441 2220 4443 2220 2220
747 9890881 (7.11.13.41).241 120 82424 240 120
748 9908921 (11.41.127).173 6020 1646 172 172
749 9920401 (17.41.331).43 840 11810 42 42
750 9995671 (7.31.73).631 45 222126 90 45
824 12327121 sq 3511.3511 1755 7024 3511 1755
# P 2^((P-1)/n) = 2^m = P.N + 1
01 341 2^((341-1)/34) = 2^10 = 341.3 + 1 2^340 = 1(341)
02 561 2^((561-1)/14) = 2^40 = 561.1959913775 + 1
03 645 2^((645-1)/23) = 2^28 = 645.416179 + 1
04 1105 2^((1105-1)/46) = 2^24 = 1105.15183 + 1
05 1387 2^((1387-1)/77) = 2^18 = 1387.189 + 1
06 1729 2^((1729-1)/48) = 2^36 = 1929.39745215 + 1
07 1905 2^((1905-1)/68) = 2^28 = 1905.140911 + 1
08 2047 2^((2047-1)/186) = 2^11 = 2047.1 + 1
09 2465 2^((2465-1)/44) = 2^56 = 2465.N + 1
10 2701 2^((2701-1)/75) = 2^36 = 2701.25442235 + 1
11 2821 2^((2821-1)/47) = 2^60 = 2821.N + 1
12 3277 2^((3277-1)/117) = 2^28 = 3277.81915 + 1
13 4033 2^((4033-1)/112) = 2^36 = 4033.17039295 + 1
14 4369 2^((4369-1)/273) = 2^16 = 4369.15 + 1
15 4371 2^((4371-1)/19) = 2^230 = 4371.N + 1
16 4681 2^((4681-1)/312) = 2^15 = 4681.7 + 1
17 5461 2^((5461-1)/390) = 2^14 = 5461.3 + 1
18 6601 2^((6601-1)/10) = 2^660 = 6601.N + 1.
Ascending powers of 2 are listed below for reference:-
a 2^a
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024
11 2048
12 4096
13 8192
14 16384
15 32768
16 65536
17 131072
18 262144
19 524288
20 1048576
21 2097152
22 4194304
23 8388608
24 16777216
25 33554432
26 67108864
27 134217728
28 268435456
29 536870912
30 1073741824
31 2147483648
32 4294967296
33 8589934592
34 17179869184
35 34359738368
36 68719476736
37 137438953472
38 274877906944
39 549755813888
40 1099511627776
41 2199023255552
42 4398046511104
43 8796093022208
44 17592186044416
45 35184372088832
46 70368744177664
47 140737488355328
48 281474976710656
49 562949953421312
50 1125899906842624
51 2251799813685248
52 4503599627370496
53 9007199254740992
54 18014398509481984
55 36028797018963968
56 72057594037927936
57 144115188075855872
58 288230376151711744
59 576460752303423488
60 1152921504606846976.