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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.