Group Parameters - phedny/Mammon GitHub Wiki
This page summarizes an investigation in group parameters.
Let n be a number. We list all the prime numbers within a range [2^(n-1), 2^n - 1], the factorization of totient p and the number of primitive roots.
The best candidate is the candidate for which the totient has the largest prime factor in the group.
Best:
3, [ <2, 1> ], 1
All:
3, [ <2, 1> ], 1
Best:
11, [ <2, 1>, <5, 1> ], 4
All:
11, [ <2, 1>, <5, 1> ], 4
13, [ <2, 2>, <3, 1> ], 4
17, [ <2, 4> ], 8
Best:
227, [ <2, 1>, <113, 1> ], 112
All:
131, [ <2, 1>, <5, 1>, <13, 1> ], 48
137, [ <2, 3>, <17, 1> ], 64
139, [ <2, 1>, <3, 1>, <23, 1> ], 44
149, [ <2, 2>, <37, 1> ], 72
151, [ <2, 1>, <3, 1>, <5, 2> ], 40
157, [ <2, 2>, <3, 1>, <13, 1> ], 48
163, [ <2, 1>, <3, 4> ], 54
167, [ <2, 1>, <83, 1> ], 82
173, [ <2, 2>, <43, 1> ], 84
179, [ <2, 1>, <89, 1> ], 88
181, [ <2, 2>, <3, 2>, <5, 1> ], 48
191, [ <2, 1>, <5, 1>, <19, 1> ], 72
193, [ <2, 6>, <3, 1> ], 64
197, [ <2, 2>, <7, 2> ], 84
199, [ <2, 1>, <3, 2>, <11, 1> ], 60
211, [ <2, 1>, <3, 1>, <5, 1>, <7, 1> ], 48
223, [ <2, 1>, <3, 1>, <37, 1> ], 72
227, [ <2, 1>, <113, 1> ], 112
229, [ <2, 2>, <3, 1>, <19, 1> ], 72
233, [ <2, 3>, <29, 1> ], 112
239, [ <2, 1>, <7, 1>, <17, 1> ], 96
241, [ <2, 4>, <3, 1>, <5, 1> ], 64
251, [ <2, 1>, <5, 3> ], 100
257, [ <2, 8> ], 128
Best:
65267, [ <2, 1>, <32633, 1> ], 32632
All:
32771, [ <2, 1>, <5, 1>, <29, 1>, <113, 1> ], 12544
32779, [ <2, 1>, <3, 3>, <607, 1> ], 10908
32783, [ <2, 1>, <37, 1>, <443, 1> ], 15912
32789, [ <2, 2>, <7, 1>, <1171, 1> ], 14040
32797, [ <2, 2>, <3, 2>, <911, 1> ], 10920
32801, [ <2, 5>, <5, 2>, <41, 1> ], 12800
32803, [ <2, 1>, <3, 1>, <7, 1>, <11, 1>, <71, 1> ], 8400
32831, [ <2, 1>, <5, 1>, <7, 2>, <67, 1> ], 11088
32833, [ <2, 6>, <3, 3>, <19, 1> ], 10368
32839, [ <2, 1>, <3, 1>, <13, 1>, <421, 1> ], 10080
32843, [ <2, 1>, <16421, 1> ], 16420
32869, [ <2, 2>, <3, 2>, <11, 1>, <83, 1> ], 9840
32887, [ <2, 1>, <3, 4>, <7, 1>, <29, 1> ], 9072
32909, [ <2, 2>, <19, 1>, <433, 1> ], 15552
32911, [ <2, 1>, <3, 1>, <5, 1>, <1097, 1> ], 8768
32917, [ <2, 2>, <3, 1>, <13, 1>, <211, 1> ], 10080
32933, [ <2, 2>, <8233, 1> ], 16464
32939, [ <2, 1>, <43, 1>, <383, 1> ], 16044
32941, [ <2, 2>, <3, 3>, <5, 1>, <61, 1> ], 8640
32957, [ <2, 2>, <7, 1>, <11, 1>, <107, 1> ], 12720
32969, [ <2, 3>, <13, 1>, <317, 1> ], 15168
32971, [ <2, 1>, <3, 1>, <5, 1>, <7, 1>, <157, 1> ], 7488
32983, [ <2, 1>, <3, 1>, <23, 1>, <239, 1> ], 10472
32987, [ <2, 1>, <16493, 1> ], 16492
32993, [ <2, 5>, <1031, 1> ], 16480
32999, [ <2, 1>, <7, 1>, <2357, 1> ], 14136
33013, [ <2, 2>, <3, 2>, <7, 1>, <131, 1> ], 9360
33023, [ <2, 1>, <11, 1>, <19, 1>, <79, 1> ], 14040
33029, [ <2, 2>, <23, 1>, <359, 1> ], 15752
33037, [ <2, 2>, <3, 1>, <2753, 1> ], 11008
33049, [ <2, 3>, <3, 5>, <17, 1> ], 10368
33053, [ <2, 2>, <8263, 1> ], 16524
33071, [ <2, 1>, <5, 1>, <3307, 1> ], 13224
33073, [ <2, 4>, <3, 1>, <13, 1>, <53, 1> ], 9984
33083, [ <2, 1>, <7, 1>, <17, 1>, <139, 1> ], 13248
33091, [ <2, 1>, <3, 1>, <5, 1>, <1103, 1> ], 8816
33107, [ <2, 1>, <16553, 1> ], 16552
33113, [ <2, 3>, <4139, 1> ], 16552
33119, [ <2, 1>, <29, 1>, <571, 1> ], 15960
33149, [ <2, 2>, <8287, 1> ], 16572
33151, [ <2, 1>, <3, 1>, <5, 2>, <13, 1>, <17, 1> ], 7680
33161, [ <2, 3>, <5, 1>, <829, 1> ], 13248
33179, [ <2, 1>, <53, 1>, <313, 1> ], 16224
33181, [ <2, 2>, <3, 1>, <5, 1>, <7, 1>, <79, 1> ], 7488
33191, [ <2, 1>, <5, 1>, <3319, 1> ], 13272
33199, [ <2, 1>, <3, 1>, <11, 1>, <503, 1> ], 10040
33203, [ <2, 1>, <13, 1>, <1277, 1> ], 15312
33211, [ <2, 1>, <3, 4>, <5, 1>, <41, 1> ], 8640
33223, [ <2, 1>, <3, 1>, <7, 2>, <113, 1> ], 9408
33247, [ <2, 1>, <3, 2>, <1847, 1> ], 11076
33287, [ <2, 1>, <11, 1>, <17, 1>, <89, 1> ], 14080
33289, [ <2, 3>, <3, 1>, <19, 1>, <73, 1> ], 10368
33301, [ <2, 2>, <3, 2>, <5, 2>, <37, 1> ], 8640
33311, [ <2, 1>, <5, 1>, <3331, 1> ], 13320
33317, [ <2, 2>, <8329, 1> ], 16656
33329, [ <2, 4>, <2083, 1> ], 16656
33331, [ <2, 1>, <3, 1>, <5, 1>, <11, 1>, <101, 1> ], 8000
33343, [ <2, 1>, <3, 1>, <5557, 1> ], 11112
33347, [ <2, 1>, <16673, 1> ], 16672
33349, [ <2, 2>, <3, 1>, <7, 1>, <397, 1> ], 9504
33353, [ <2, 3>, <11, 1>, <379, 1> ], 15120
33359, [ <2, 1>, <13, 1>, <1283, 1> ], 15384
33377, [ <2, 5>, <7, 1>, <149, 1> ], 14208
33391, [ <2, 1>, <3, 2>, <5, 1>, <7, 1>, <53, 1> ], 7488
33403, [ <2, 1>, <3, 1>, <19, 1>, <293, 1> ], 10512
33409, [ <2, 7>, <3, 2>, <29, 1> ], 10752
33413, [ <2, 2>, <8353, 1> ], 16704
33427, [ <2, 1>, <3, 3>, <619, 1> ], 11124
33457, [ <2, 4>, <3, 1>, <17, 1>, <41, 1> ], 10240
33461, [ <2, 2>, <5, 1>, <7, 1>, <239, 1> ], 11424
33469, [ <2, 2>, <3, 1>, <2789, 1> ], 11152
33479, [ <2, 1>, <19, 1>, <881, 1> ], 15840
33487, [ <2, 1>, <3, 1>, <5581, 1> ], 11160
33493, [ <2, 2>, <3, 1>, <2791, 1> ], 11160
33503, [ <2, 1>, <7, 1>, <2393, 1> ], 14352
33521, [ <2, 4>, <5, 1>, <419, 1> ], 13376
33529, [ <2, 3>, <3, 1>, <11, 1>, <127, 1> ], 10080
33533, [ <2, 2>, <83, 1>, <101, 1> ], 16400
33547, [ <2, 1>, <3, 1>, <5591, 1> ], 11180
33563, [ <2, 1>, <97, 1>, <173, 1> ], 16512
33569, [ <2, 5>, <1049, 1> ], 16768
33577, [ <2, 3>, <3, 1>, <1399, 1> ], 11184
33581, [ <2, 2>, <5, 1>, <23, 1>, <73, 1> ], 12672
33587, [ <2, 1>, <7, 1>, <2399, 1> ], 14388
33589, [ <2, 2>, <3, 3>, <311, 1> ], 11160
33599, [ <2, 1>, <107, 1>, <157, 1> ], 16536
33601, [ <2, 6>, <3, 1>, <5, 2>, <7, 1> ], 7680
33613, [ <2, 2>, <3, 1>, <2801, 1> ], 11200
33617, [ <2, 4>, <11, 1>, <191, 1> ], 15200
33619, [ <2, 1>, <3, 1>, <13, 1>, <431, 1> ], 10320
33623, [ <2, 1>, <16811, 1> ], 16810
33629, [ <2, 2>, <7, 1>, <1201, 1> ], 14400
33637, [ <2, 2>, <3, 1>, <2803, 1> ], 11208
33641, [ <2, 3>, <5, 1>, <29, 2> ], 12992
33647, [ <2, 1>, <16823, 1> ], 16822
33679, [ <2, 1>, <3, 2>, <1871, 1> ], 11220
33703, [ <2, 1>, <3, 1>, <41, 1>, <137, 1> ], 10880
33713, [ <2, 4>, <7, 2>, <43, 1> ], 14112
33721, [ <2, 3>, <3, 1>, <5, 1>, <281, 1> ], 8960
33739, [ <2, 1>, <3, 1>, <5623, 1> ], 11244
33749, [ <2, 2>, <11, 1>, <13, 1>, <59, 1> ], 13920
33751, [ <2, 1>, <3, 3>, <5, 4> ], 9000
33757, [ <2, 2>, <3, 1>, <29, 1>, <97, 1> ], 10752
33767, [ <2, 1>, <16883, 1> ], 16882
33769, [ <2, 3>, <3, 2>, <7, 1>, <67, 1> ], 9504
33773, [ <2, 2>, <8443, 1> ], 16884
33791, [ <2, 1>, <5, 1>, <31, 1>, <109, 1> ], 12960
33797, [ <2, 2>, <7, 1>, <17, 1>, <71, 1> ], 13440
33809, [ <2, 4>, <2113, 1> ], 16896
33811, [ <2, 1>, <3, 1>, <5, 1>, <7, 2>, <23, 1> ], 7392
33827, [ <2, 1>, <13, 1>, <1301, 1> ], 15600
33829, [ <2, 2>, <3, 1>, <2819, 1> ], 11272
33851, [ <2, 1>, <5, 2>, <677, 1> ], 13520
33857, [ <2, 6>, <23, 2> ], 16192
33863, [ <2, 1>, <16931, 1> ], 16930
33871, [ <2, 1>, <3, 1>, <5, 1>, <1129, 1> ], 9024
33889, [ <2, 5>, <3, 1>, <353, 1> ], 11264
33893, [ <2, 2>, <37, 1>, <229, 1> ], 16416
33911, [ <2, 1>, <5, 1>, <3391, 1> ], 13560
33923, [ <2, 1>, <7, 1>, <2423, 1> ], 14532
33931, [ <2, 1>, <3, 2>, <5, 1>, <13, 1>, <29, 1> ], 8064
33937, [ <2, 4>, <3, 1>, <7, 1>, <101, 1> ], 9600
33941, [ <2, 2>, <5, 1>, <1697, 1> ], 13568
33961, [ <2, 3>, <3, 1>, <5, 1>, <283, 1> ], 9024
33967, [ <2, 1>, <3, 3>, <17, 1>, <37, 1> ], 10368
33997, [ <2, 2>, <3, 1>, <2833, 1> ], 11328
34019, [ <2, 1>, <73, 1>, <233, 1> ], 16704
34031, [ <2, 1>, <5, 1>, <41, 1>, <83, 1> ], 13120
34033, [ <2, 4>, <3, 1>, <709, 1> ], 11328
34039, [ <2, 1>, <3, 2>, <31, 1>, <61, 1> ], 10800
34057, [ <2, 3>, <3, 2>, <11, 1>, <43, 1> ], 10080
34061, [ <2, 2>, <5, 1>, <13, 1>, <131, 1> ], 12480
34123, [ <2, 1>, <3, 1>, <11, 2>, <47, 1> ], 10120
34127, [ <2, 1>, <113, 1>, <151, 1> ], 16800
34129, [ <2, 4>, <3, 3>, <79, 1> ], 11232
34141, [ <2, 2>, <3, 1>, <5, 1>, <569, 1> ], 9088
34147, [ <2, 1>, <3, 2>, <7, 1>, <271, 1> ], 9720
34157, [ <2, 2>, <8539, 1> ], 17076
34159, [ <2, 1>, <3, 1>, <5693, 1> ], 11384
34171, [ <2, 1>, <3, 1>, <5, 1>, <17, 1>, <67, 1> ], 8448
34183, [ <2, 1>, <3, 4>, <211, 1> ], 11340
34211, [ <2, 1>, <5, 1>, <11, 1>, <311, 1> ], 12400
34213, [ <2, 2>, <3, 1>, <2851, 1> ], 11400
34217, [ <2, 3>, <7, 1>, <13, 1>, <47, 1> ], 13248
34231, [ <2, 1>, <3, 1>, <5, 1>, <7, 1>, <163, 1> ], 7776
34253, [ <2, 2>, <8563, 1> ], 17124
34259, [ <2, 1>, <7, 1>, <2447, 1> ], 14676
34261, [ <2, 2>, <3, 1>, <5, 1>, <571, 1> ], 9120
34267, [ <2, 1>, <3, 1>, <5711, 1> ], 11420
34273, [ <2, 5>, <3, 2>, <7, 1>, <17, 1> ], 9216
34283, [ <2, 1>, <61, 1>, <281, 1> ], 16800
34297, [ <2, 3>, <3, 1>, <1429, 1> ], 11424
34301, [ <2, 2>, <5, 2>, <7, 3> ], 11760
34303, [ <2, 1>, <3, 1>, <5717, 1> ], 11432
34313, [ <2, 3>, <4289, 1> ], 17152
34319, [ <2, 1>, <17159, 1> ], 17158
34327, [ <2, 1>, <3, 2>, <1907, 1> ], 11436
34337, [ <2, 5>, <29, 1>, <37, 1> ], 16128
34351, [ <2, 1>, <3, 1>, <5, 2>, <229, 1> ], 9120
34361, [ <2, 3>, <5, 1>, <859, 1> ], 13728
34367, [ <2, 1>, <17183, 1> ], 17182
34369, [ <2, 6>, <3, 1>, <179, 1> ], 11392
34381, [ <2, 2>, <3, 2>, <5, 1>, <191, 1> ], 9120
34403, [ <2, 1>, <103, 1>, <167, 1> ], 16932
34421, [ <2, 2>, <5, 1>, <1721, 1> ], 13760
34429, [ <2, 2>, <3, 1>, <19, 1>, <151, 1> ], 10800
34439, [ <2, 1>, <67, 1>, <257, 1> ], 16896
34457, [ <2, 3>, <59, 1>, <73, 1> ], 16704
34469, [ <2, 2>, <7, 1>, <1231, 1> ], 14760
34471, [ <2, 1>, <3, 2>, <5, 1>, <383, 1> ], 9168
34483, [ <2, 1>, <3, 1>, <7, 1>, <821, 1> ], 9840
34487, [ <2, 1>, <43, 1>, <401, 1> ], 16800
34499, [ <2, 1>, <47, 1>, <367, 1> ], 16836
34501, [ <2, 2>, <3, 1>, <5, 3>, <23, 1> ], 8800
34511, [ <2, 1>, <5, 1>, <7, 1>, <17, 1>, <29, 1> ], 10752
34513, [ <2, 4>, <3, 1>, <719, 1> ], 11488
34519, [ <2, 1>, <3, 1>, <11, 1>, <523, 1> ], 10440
34537, [ <2, 3>, <3, 1>, <1439, 1> ], 11504
34543, [ <2, 1>, <3, 2>, <19, 1>, <101, 1> ], 10800
34549, [ <2, 2>, <3, 1>, <2879, 1> ], 11512
34583, [ <2, 1>, <17291, 1> ], 17290
34589, [ <2, 2>, <8647, 1> ], 17292
34591, [ <2, 1>, <3, 1>, <5, 1>, <1153, 1> ], 9216
34603, [ <2, 1>, <3, 1>, <73, 1>, <79, 1> ], 11232
34607, [ <2, 1>, <11, 3>, <13, 1> ], 14520
34613, [ <2, 2>, <17, 1>, <509, 1> ], 16256
34631, [ <2, 1>, <5, 1>, <3463, 1> ], 13848
34649, [ <2, 3>, <61, 1>, <71, 1> ], 16800
34651, [ <2, 1>, <3, 2>, <5, 2>, <7, 1>, <11, 1> ], 7200
34667, [ <2, 1>, <17333, 1> ], 17332
34673, [ <2, 4>, <11, 1>, <197, 1> ], 15680
34679, [ <2, 1>, <7, 1>, <2477, 1> ], 14856
34687, [ <2, 1>, <3, 2>, <41, 1>, <47, 1> ], 11040
34693, [ <2, 2>, <3, 1>, <7, 2>, <59, 1> ], 9744
34703, [ <2, 1>, <17351, 1> ], 17350
34721, [ <2, 5>, <5, 1>, <7, 1>, <31, 1> ], 11520
34729, [ <2, 3>, <3, 1>, <1447, 1> ], 11568
34739, [ <2, 1>, <11, 1>, <1579, 1> ], 15780
34747, [ <2, 1>, <3, 1>, <5791, 1> ], 11580
34757, [ <2, 2>, <8689, 1> ], 17376
34759, [ <2, 1>, <3, 2>, <1931, 1> ], 11580
34763, [ <2, 1>, <7, 1>, <13, 1>, <191, 1> ], 13680
34781, [ <2, 2>, <5, 1>, <37, 1>, <47, 1> ], 13248
34807, [ <2, 1>, <3, 1>, <5801, 1> ], 11600
34819, [ <2, 1>, <3, 1>, <7, 1>, <829, 1> ], 9936
34841, [ <2, 3>, <5, 1>, <13, 1>, <67, 1> ], 12672
34843, [ <2, 1>, <3, 1>, <5807, 1> ], 11612
34847, [ <2, 1>, <7, 1>, <19, 1>, <131, 1> ], 14040
34849, [ <2, 5>, <3, 2>, <11, 2> ], 10560
34871, [ <2, 1>, <5, 1>, <11, 1>, <317, 1> ], 12640
34877, [ <2, 2>, <8719, 1> ], 17436
34883, [ <2, 1>, <107, 1>, <163, 1> ], 17172
34897, [ <2, 4>, <3, 1>, <727, 1> ], 11616
34913, [ <2, 5>, <1091, 1> ], 17440
34919, [ <2, 1>, <13, 1>, <17, 1>, <79, 1> ], 14976
34939, [ <2, 1>, <3, 3>, <647, 1> ], 11628
34949, [ <2, 2>, <8737, 1> ], 17472
34961, [ <2, 4>, <5, 1>, <19, 1>, <23, 1> ], 12672
34963, [ <2, 1>, <3, 1>, <5827, 1> ], 11652
34981, [ <2, 2>, <3, 1>, <5, 1>, <11, 1>, <53, 1> ], 8320
35023, [ <2, 1>, <3, 1>, <13, 1>, <449, 1> ], 10752
35027, [ <2, 1>, <83, 1>, <211, 1> ], 17220
35051, [ <2, 1>, <5, 2>, <701, 1> ], 14000
35053, [ <2, 2>, <3, 1>, <23, 1>, <127, 1> ], 11088
35059, [ <2, 1>, <3, 1>, <5843, 1> ], 11684
35069, [ <2, 2>, <11, 1>, <797, 1> ], 15920
35081, [ <2, 3>, <5, 1>, <877, 1> ], 14016
35083, [ <2, 1>, <3, 2>, <1949, 1> ], 11688
35089, [ <2, 4>, <3, 1>, <17, 1>, <43, 1> ], 10752
35099, [ <2, 1>, <7, 1>, <23, 1>, <109, 1> ], 14256
35107, [ <2, 1>, <3, 1>, <5851, 1> ], 11700
35111, [ <2, 1>, <5, 1>, <3511, 1> ], 14040
35117, [ <2, 2>, <8779, 1> ], 17556
35129, [ <2, 3>, <4391, 1> ], 17560
35141, [ <2, 2>, <5, 1>, <7, 1>, <251, 1> ], 12000
35149, [ <2, 2>, <3, 1>, <29, 1>, <101, 1> ], 11200
35153, [ <2, 4>, <13, 3> ], 16224
35159, [ <2, 1>, <17579, 1> ], 17578
35171, [ <2, 1>, <5, 1>, <3517, 1> ], 14064
35201, [ <2, 7>, <5, 2>, <11, 1> ], 12800
35221, [ <2, 2>, <3, 1>, <5, 1>, <587, 1> ], 9376
35227, [ <2, 1>, <3, 2>, <19, 1>, <103, 1> ], 11016
35251, [ <2, 1>, <3, 1>, <5, 3>, <47, 1> ], 9200
35257, [ <2, 3>, <3, 1>, <13, 1>, <113, 1> ], 10752
35267, [ <2, 1>, <7, 1>, <11, 1>, <229, 1> ], 13680
35279, [ <2, 1>, <31, 1>, <569, 1> ], 17040
35281, [ <2, 4>, <3, 2>, <5, 1>, <7, 2> ], 8064
35291, [ <2, 1>, <5, 1>, <3529, 1> ], 14112
35311, [ <2, 1>, <3, 1>, <5, 1>, <11, 1>, <107, 1> ], 8480
35317, [ <2, 2>, <3, 4>, <109, 1> ], 11664
35323, [ <2, 1>, <3, 1>, <7, 1>, <29, 2> ], 9744
35327, [ <2, 1>, <17, 1>, <1039, 1> ], 16608
35339, [ <2, 1>, <17669, 1> ], 17668
35353, [ <2, 3>, <3, 2>, <491, 1> ], 11760
35363, [ <2, 1>, <17681, 1> ], 17680
35381, [ <2, 2>, <5, 1>, <29, 1>, <61, 1> ], 13440
35393, [ <2, 6>, <7, 1>, <79, 1> ], 14976
35401, [ <2, 3>, <3, 1>, <5, 2>, <59, 1> ], 9280
35407, [ <2, 1>, <3, 2>, <7, 1>, <281, 1> ], 10080
35419, [ <2, 1>, <3, 1>, <5903, 1> ], 11804
35423, [ <2, 1>, <89, 1>, <199, 1> ], 17424
35437, [ <2, 2>, <3, 1>, <2953, 1> ], 11808
35447, [ <2, 1>, <37, 1>, <479, 1> ], 17208
35449, [ <2, 3>, <3, 1>, <7, 1>, <211, 1> ], 10080
35461, [ <2, 2>, <3, 2>, <5, 1>, <197, 1> ], 9408
35491, [ <2, 1>, <3, 1>, <5, 1>, <7, 1>, <13, 2> ], 7488
35507, [ <2, 1>, <41, 1>, <433, 1> ], 17280
35509, [ <2, 2>, <3, 1>, <11, 1>, <269, 1> ], 10720
35521, [ <2, 6>, <3, 1>, <5, 1>, <37, 1> ], 9216
35527, [ <2, 1>, <3, 1>, <31, 1>, <191, 1> ], 11400
35531, [ <2, 1>, <5, 1>, <11, 1>, <17, 1>, <19, 1> ], 11520
35533, [ <2, 2>, <3, 3>, <7, 1>, <47, 1> ], 9936
35537, [ <2, 4>, <2221, 1> ], 17760
35543, [ <2, 1>, <13, 1>, <1367, 1> ], 16392
35569, [ <2, 4>, <3, 2>, <13, 1>, <19, 1> ], 10368
35573, [ <2, 2>, <8893, 1> ], 17784
35591, [ <2, 1>, <5, 1>, <3559, 1> ], 14232
35593, [ <2, 3>, <3, 1>, <1483, 1> ], 11856
35597, [ <2, 2>, <11, 1>, <809, 1> ], 16160
35603, [ <2, 1>, <7, 1>, <2543, 1> ], 15252
35617, [ <2, 5>, <3, 1>, <7, 1>, <53, 1> ], 9984
35671, [ <2, 1>, <3, 1>, <5, 1>, <29, 1>, <41, 1> ], 8960
35677, [ <2, 2>, <3, 2>, <991, 1> ], 11880
35729, [ <2, 4>, <7, 1>, <11, 1>, <29, 1> ], 13440
35731, [ <2, 1>, <3, 2>, <5, 1>, <397, 1> ], 9504
35747, [ <2, 1>, <61, 1>, <293, 1> ], 17520
35753, [ <2, 3>, <41, 1>, <109, 1> ], 17280
35759, [ <2, 1>, <19, 1>, <941, 1> ], 16920
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65419, [ <2, 1>, <3, 1>, <10903, 1> ], 21804
65423, [ <2, 1>, <7, 1>, <4673, 1> ], 28032
65437, [ <2, 2>, <3, 1>, <7, 1>, <19, 1>, <41, 1> ], 17280
65447, [ <2, 1>, <43, 1>, <761, 1> ], 31920
65449, [ <2, 3>, <3, 4>, <101, 1> ], 21600
65479, [ <2, 1>, <3, 1>, <7, 1>, <1559, 1> ], 18696
65497, [ <2, 3>, <3, 1>, <2729, 1> ], 21824
65519, [ <2, 1>, <17, 1>, <41, 1>, <47, 1> ], 29440
65521, [ <2, 4>, <3, 2>, <5, 1>, <7, 1>, <13, 1> ], 13824
65537, [ <2, 16> ], 32768
No exact results are know. Use the gist 2349984 at the Magma Calculator to improve the best known candidates.
Looking into the the last few best know bounds the have following form. (p = 2q + 1, q) with both p and q prime. These kinds of primes are known as Sophie Germain primes. The largest of the two primes is known as a safe prime. Gist 2376358 exploits this fact and can be used to improve the candidates below.
The below list will be restricted to Sophie Germain primes.
32-bit: 4294944443, [ <2, 1>, <2147472221, 1> ], 2147472220
64-bit: 18401784977709140003, [ <2, 1>, <9200892488854570001, 1> ] 9200892488854570000
128-bit: 338920273030085779859273003603875667999, [ <2, 1>, <169460136515042889929636501801937833999, 1> ], 169460136515042889929636501801937833998
256-bit: 88956204038081157586553139672739952408919742301864736598576051731048383783999, [ <2, 1>, <44478102019040578793276569836369976204459871150932368299288025865524191891999, 1> ], 44478102019040578793276569836369976204459871150932368299288025865524191891998