[seqfan] Glued primes

Artur grafix at csl.pl
Sun Nov 23 17:49:13 CET 2008


Dear Max,

I was check few samples
e.g. pair of primes {10429407431911334611, 918125051602568899753}
which are two the biggest prime divisors of 2^243+1 and 2^486-1
486=2*243
occured every time in pairs (never separately) and both in Fermat as in 
Mersenne numbers (infinite many times)
(I know few more such pairs) and many many products of such pairs but I 
can't factorized these on separate primes because I was mentioned 
yesterday these cases are most difficult to factorization in anyone method.
I don't know yet that triples and more primes product occured but I'm 
examined these now.

exponents for Mersenne numbers are

{486, 972, 1458, 1944, 2430, 2916, 3402, 3888, 4374, 4860, 5346, \
5832, 6318, 6804, 7290, 7776, 8262, 8748, 9234, 9720, 10206, 10692, \
11178, 11664, 12150, 12636, 13122, 13608, 14094, 14580, 15066, 15552, \
16038, 16524, 17010, 17496, 17982, 18468, 18954, 19440, 19926, 20412, \
20898, 21384, 21870, 22356, 22842, 23328, 23814, 24300, 24786, 25272, \
25758, 26244, 26730, 27216, 27702, 28188, 28674, 29160, 29646, 30132, \
30618, 31104, 31590, 32076, 32562, 33048, 33534, 34020, 34506, 34992, \
35478, 35964, 36450, 36936, 37422, 37908, 38394, 38880, 39366, 39852, \
40338, 40824, 41310, 41796, 42282, 42768, 43254, 43740, 44226, 44712, \
45198, 45684, 46170, 46656, 47142, 47628, 48114, 48600, 49086, 49572, \
50058, 50544, 51030, 51516, 52002, 52488, 52974, 53460, 53946, 54432, \
54918, 55404, 55890, 56376, 56862, 57348, 57834, 58320, 58806, 59292, \
59778, 60264, 60750, 61236, 61722, 62208, 62694, 63180, 63666, 64152, \
64638, 65124, 65610, 66096, 66582, 67068, 67554, 68040, 68526, 69012, \
69498, 69984, 70470, 70956, 71442, 71928, 72414, 72900, 73386, 73872, \
74358, 74844, 75330, 75816, 76302, 76788, 77274, 77760, 78246, 78732, \
79218, 79704, 80190, 80676, 81162, 81648, 82134, 82620, 83106, 83592, \
84078, 84564, 85050, 85536, 86022, 86508, 86994, 87480, 87966, 88452, \
88938, 89424, 89910, 90396, 90882, 91368, 91854, 92340, 92826, 93312, \
93798, 94284, 94770, 95256, 95742, 96228, 96714, 97200, 97686, 98172, \
98658, 99144, 99630}

Exponenents for Fermat numbers are

{243, 729, 1215, 1701, 2187, 2673, 3159, 3645, 4131, 4617, 5103, \
5589, 6075, 6561, 7047, 7533, 8019, 8505, 8991, 9477, 9963, 10449, \
10935, 11421, 11907, 12393, 12879, 13365, 13851, 14337, 14823, 15309, \
15795, 16281, 16767, 17253, 17739, 18225, 18711, 19197, 19683, 20169, \
20655, 21141, 21627, 22113, 22599, 23085, 23571, 24057, 24543, 25029, \
25515, 26001, 26487, 26973, 27459, 27945, 28431, 28917, 29403, 29889, \
30375, 30861, 31347, 31833, 32319, 32805, 33291, 33777, 34263, 34749, \
35235, 35721, 36207, 36693, 37179, 37665, 38151, 38637, 39123, 39609, \
40095, 40581, 41067, 41553, 42039, 42525, 43011, 43497, 43983, 44469, \
44955, 45441, 45927, 46413, 46899, 47385, 47871, 48357, 48843, 49329, \
49815, 50301, 50787, 51273, 51759, 52245, 52731, 53217, 53703, 54189, \
54675, 55161, 55647, 56133, 56619, 57105, 57591, 58077, 58563, 59049, \
59535, 60021, 60507, 60993, 61479, 61965, 62451, 62937, 63423, 63909, \
64395, 64881, 65367, 65853, 66339, 66825, 67311, 67797, 68283, 68769, \
69255, 69741, 70227, 70713, 71199, 71685, 72171, 72657, 73143, 73629, \
74115, 74601, 75087, 75573, 76059, 76545, 77031, 77517, 78003, 78489, \
78975, 79461, 79947, 80433, 80919, 81405, 81891, 82377, 82863, 83349, \
83835, 84321, 84807, 85293, 85779, 86265, 86751, 87237, 87723, 88209, \
88695, 89181, 89667, 90153, 90639, 91125, 91611, 92097, 92583, 93069, \
93555, 94041, 94527, 95013, 95499, 95985, 96471, 96957, 97443, 97929, \
98415, 98901, 99387, 99873}




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