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Leonhard
Euler showed:
Theorem : If k >1 and
p =4k +3 is prime, then 2p +1 is prime if and
only if 2p +1 divides 2p p is an odd composite, then 2p p The
new Mersenne conjecture , Amer. Math. Monthly, 96 (1989) 125-128 [BSW89 ])
have conjectured the following.
Conjecture (* ):
Let p be any odd natural number. If two of the following
conditions hold, then so does the third:
p = 2k p = 4k
2p Mersenne
prime )
(2p
Below we list all of the primes p where
p < 20000000 and p = 2k p = 4k 2p
(2p probable-prime .
Details are in the table and notes below.
(The conjecture is that if any two of the entries in a row is yes, so
is the third.)
p p = 2k k 2p
(2p
3
yes (-1)
yes
yes
5
yes (+1)
yes
yes
7
yes (-1 or +3)
yes
yes
11
no
no: 23
yes
13
yes (-3)
yes
yes
17
yes (+1)
yes
yes
19
yes (+3)
yes
yes
23
no
no: 47
yes
31
yes (-1)
yes
yes
43
no
no: 431
yes
61
yes (-3)
yes
yes
67
yes (+3)
no: 193707721
no: 7327657
79
no
no: 2867
yes
89
no
yes
no: 179
101
no
no: 7432339208719
yes
107
no
yes
no: 643
127
yes (-1)
yes
yes
167
no
no: 2349023
yes
191
no
no: 383
yes
199
no
no: 164504919713
yes
257
yes (+1)
no: 535006138814359
no: 37239639534523
313
no
no: 10960009
yes
347
no
no: 14143189112952632419639
yes
521
no
yes
no: 501203
607
no
yes
no: 115331
701
no
no: 796337
yes
1021
yes (-3)
no: 40841
no: 10211
1279
no
yes
no: 706009
1709
no
no: 379399
yes [Morain1990a ]
2203
no
yes
no: 13219
2281
no
yes
no: 22811
2617
no
no: 78511
yes [Morain1990a ]
3217
no
yes
no: 7489177
3539
no
no: 7079
yes [Morain1990a ]
4093
yes (-3)
no: 2397911088359
no: 3732912210059
4099
yes(+3)
no: 73783
no: 2164273
4253
no
yes
no: 118071787
4423
no
yes
no: 2827782322058633
5807
no
no: 139369
yes (note 6 )
8191
yes (-1)
no: 338193759479
no
9689
no
yes
no: 19379
9941
no
yes
no: 11120148512909357034073
10501
no
no: 2160708549249199
yes (note 5 )
10691
no
no: 21383
yes (note 1 )
11213
no
yes
no
11279
no
no: 2198029886879
yes (note 4 )
12391
no
no: 198257
yes (note 3 )
14479
no
no: 27885728233673
yes (note 2 )
16381
yes (-3)
no
no: 163811
19937
no
yes
no
21701
no
yes
no: 43403
23209
no
yes
no: 4688219
42737
no
no
yes (xxx)
yes (note 7 )
44497
no
yes
no: 2135857
65537
yes (+1)
no
no: 13091975735977
65539
yes (+3)
no: 3354489977369
no: 58599599603
83339
no
no: 166679
yes (prp)
86243
no
yes
no: 1627710365249
95369
no
no: 297995890279
yes (prp)
110503
no
yes
no
117239
no
no
yes (prp)
127031
no
no: 12194977
yes (prp)
131071
yes (-1)
no: 231733529
no: 2883563
132049
no
yes
no: 618913299601153
138937
no
no: 100068818503
yes (prp)
141079
no
no: 458506751
yes (prp)
216091
no
yes
no
262147
yes (+3)
no: 268179002471
no: 4194353
267017
no
no: 1602103
yes (prp)
269987
no
no
yes (prp)
374321
no
no
yes (prp)
524287
yes (-1)
no: 62914441
no
756839
no
yes
no: 1640826953
859433
no
yes
no: 1718867
1048573
yes (-3)
no: 73400111
no
1257787
no
yes
no: 20124593
1398269
no
yes
no
2976221
no
yes
no: 434313978089
3021377
no
yes
no: 95264016811
4194301
yes (-3)
no: 2873888432993463577
no: 14294177809
6972593
no
yes
no
13466917
no
yes
no: 781081187
16777213
yes (-3)
no
unknown (no factor <10^17)
20996011
no
yes
unknown (no factor <10^18)
24036583
no
yes
no: 11681779339
25964951
no
yes
no: 155789707
30402457
no
yes
unknown (no factor < 10^18)
When a small factor is known we listed it above.
Notes:
##
prime
note
*
(any)
The integers listed after 'no' are small factors of the corresponding
composite.
**
(any)
The expression prp means probable-prime
1
10691
ECPP primality proof by David Broadhurst via Primo, certificate
n10691.zip
2
14479
ECPP primality proof by David Broadhurst via Primo, certificate
n14479.zip
3
12391
Proof by François Morain, see his notes .
4
11279
Proof by Preda Mihailescu, see his notes .
5
10501
Proof by François Morain, see his notes .
6
5807
Proof by Preda Mihailescu, see his notes .
7
42737
Proof by François Morain, see his notes
on the prime's page.
