Right Perfect Prime Numbers

Housemouse · 2008-05-27, 14:18

If P is an even perfect number greater than 6, P-1 is always composite divisible by nine. Is it known which perfect numbers are prime for P+1?

R.D. Silverman · 2008-05-27, 14:36

Quote:

Originally Posted by Housemouse

If P is an even perfect number greater than 6, P-1 is always composite divisible by nine. Is it known which perfect numbers are prime for P+1?

Clearly not. We don't even know whether the Mersenne primes are
infinite in number.

R. Gerbicz · 2008-05-27, 15:45

Quote:

Originally Posted by Housemouse

If P is an even perfect number greater than 6, P-1 is always composite divisible by nine. Is it known which perfect numbers are prime for P+1?

http://www.research.att.com/~njas/sequences/A061644

petrw1 · 2008-05-27, 16:07

6 (7)
29 (29)
33550336 (33550337)
are Prime

496 (497)
8128 (8129)
8589869056 (8589869057)
are Composite

That is as far as I checked

ATH · 2008-05-28, 11:01

I trialfactored P+1 for the 44 known perfect numbers P and did ECM on 1 of them:

p: factor(s) of 2^p-1*(2^p-1) + 1
2: prime
3: prime
5: 7,71
7: 11,739
13: prime
17: 7,11,111556741
19: prime
31: 29,71,137,1621,5042777503
61: 2432582681,1092853292237112554142488617
89: 7
107: 7,11,67
127: 11,107,261697
521: 7,71
607: 11
1279: 72353441721527140856665601867
2203: 60449,1498429,711309659
2281: 197,557,1999,92033
3217: 11
4253: 7,53,8731,2353129,50820071
4423: 2163571
9689: 7,211,49922567
9941: 7,67,1605697,194147011
11213: 7
19937: 7,11,1129,168457
21701: 7
23209: 35603,620377
44497: 11,13259,16177141,896297147
86243: 7,29,301123,26072029
110503: 491,1493,1529761
132049: ?
216091: 4673,6920341
756839: 7
859433: 7
1257787: 11
1398269: 7,53,12713,17425081,199979189
2976221: 7,71
3021377: 7,11,49603
6972593: 7,6007,8392897,52193821
13466917: 11,45007
20996011: 1552147,114242767
24036583: 149
25964951: 7
30402457: 11
32582657: 7,11,67,34549,127541

So perfectnumber+1 are prime for p=2,3,13 and 19 and unknown for p=132049 (79502 digits) which I trialfactored to 18*10⁹.

R. Gerbicz · 2008-05-28, 11:22

Quote:

Originally Posted by ATH

So perfectnumber+1 are prime for p=2,3,13 and 19 and unknown for p=132049 (79502 digits) which I trialfactored to 18*10⁹.

Please note that if N=2^(p-1)*(2^p-1)+1 (where Mp=2^p-1 is a Mersenne prime), then the primefactorization of N-1 is known so a quick exact primetest is possible.

henryzz · 2008-05-29, 08:25

Quote:

Originally Posted by ATH

I trialfactored P+1 for the 44 known perfect numbers P and did ECM on 1 of them:

p: factor(s) of 2^p-1*(2^p-1) + 1
2: prime
3: prime
5: 7,71
7: 11,739
13: prime
17: 7,11,111556741
19: prime
31: 29,71,137,1621,5042777503
61: 2432582681,1092853292237112554142488617
89: 7
107: 7,11,67
127: 11,107,261697
521: 7,71
607: 11
1279: 72353441721527140856665601867
2203: 60449,1498429,711309659
2281: 197,557,1999,92033
3217: 11
4253: 7,53,8731,2353129,50820071
4423: 2163571
9689: 7,211,49922567
9941: 7,67,1605697,194147011
11213: 7
19937: 7,11,1129,168457
21701: 7
23209: 35603,620377
44497: 11,13259,16177141,896297147
86243: 7,29,301123,26072029
110503: 491,1493,1529761
132049: ?
216091: 4673,6920341
756839: 7
859433: 7
1257787: 11
1398269: 7,53,12713,17425081,199979189
2976221: 7,71
3021377: 7,11,49603
6972593: 7,6007,8392897,52193821
13466917: 11,45007
20996011: 1552147,114242767
24036583: 149
25964951: 7
30402457: 11
32582657: 7,11,67,34549,127541

So perfectnumber+1 are prime for p=2,3,13 and 19 and unknown for p=132049 (79502 digits) which I trialfactored to 18*10⁹.

gmp-ecm doesnt think p=132049 is prp

philmoore · 2008-05-29, 16:59

Quote:

Originally Posted by henryzz

gmp-ecm doesnt think p=132049 is prp

If you follow the link given at the site given by R. Gerbicz,
http://www.primepuzzles.net/puzzles/puzz_203.htm ,
you will see that PrimeForm agrees with gmp-ecm on this.

ATH · 2008-05-30, 00:30

Question solved.

I found a factor of 2^p-1*(2^p-1) + 1 for p=132049 with gmp-ecm:
194528547122653

So of the 44 known perfect numbers P=2^p-1*(2^p-1), P+1 is only prime for p=2,3,13 and 19.

ATH · 2008-09-16, 23:19

Updated list:

Code:

p:		factor(s) of 2^p-1*(2^p-1) + 1
------------------------------------------------------------------
2:		prime
3:		prime
5:		7 , 71
7:		11 , 739
13:		prime
17:		7 , 11 , 111556741
19:		prime
31:		29 , 71 , 137 , 1621 , 5042777503
61:		2432582681 , 1092853292237112554142488617
89:		7
107:		7 , 11 , 67
127:		11 , 107 , 261697
521:		7 , 71
607:		11
1279:		72353441721527140856665601867
2203:		60449 , 1498429 , 711309659, 1418050069
2281:		197 , 557 , 1999 , 92033
3217:		11
4253:		7 , 53 , 8731 , 2353129 , 50820071
4423:		2163571
9689:		7 , 211 , 49922567
9941:		7 , 67 , 1605697 , 194147011
11213:		7
19937:		7 , 11 , 1129 , 168457
21701:		7
23209:		35603 , 620377
44497:		11 , 13259 , 16177141 , 896297147
86243:		7 , 29 , 301123 , 26072029
110503:		491 , 1493 , 1529761
132049:		194528547122653
216091:		4673 , 6920341
756839:		7
859433:		7
1257787:	11
1398269:	7 , 53 , 12713 , 17425081 , 199979189
2976221:	7 , 71
3021377:	7 , 11 , 49603
6972593:	7 , 6007 , 8392897 , 52193821
13466917:	11 , 45007
20996011:	1552147 , 114242767
24036583:	149
25964951:	7
30402457:	11
32582657:	7 , 11 , 67 , 34549 , 127541
37156667:	7 , 11 , 44753 , 202577 , 1282451377
42643801:	3593 , 7089208037
43112609:	7 , 211 , 70121 , 71647 , 1846524311
57885161:	7 , 22127627
74207281:	No factor < 6*10¹²
77232917:	7 , 11 , 11587
82589933:	7 , 67 , 599 , 7347113 , 14416229

So of the now 51 known perfect numbers P=2^p-1*(2^p-1), P+1 is only still prime for p=2,3,13 and 19, and status unknown for p=74207281.

JeppeSN · 2016-01-23, 09:38

Can we extend this?

p: factor(s) of 2p-1*(2p-1) + 1
57885161: 7
74207281: ?

/JeppeSN

2008-05-27, 14:18	#1
Housemouse Feb 2008 2⁵ Posts	Right Perfect Prime Numbers If P is an even perfect number greater than 6, P-1 is always composite divisible by nine. Is it known which perfect numbers are prime for P+1?

2008-05-27, 16:07	#4
petrw1 1976 Toyota Corona years forever! "Wayne" Nov 2006 Saskatchewan, Canada 43·107 Posts	yes 6 (7) 29 (29) 33550336 (33550337) are Prime 496 (497) 8128 (8129) 8589869056 (8589869057) are Composite That is as far as I checked

2008-05-28, 11:01	#5
ATH Einyen Dec 2003 Denmark 3×5²×41 Posts	I trialfactored P+1 for the 44 known perfect numbers P and did ECM on 1 of them: p: factor(s) of 2^p-1(2^p-1) + 1 2: prime 3: prime 5: 7,71 7: 11,739 13: prime 17: 7,11,111556741 19: prime 31: 29,71,137,1621,5042777503 61: 2432582681,1092853292237112554142488617 89: 7 107: 7,11,67 127: 11,107,261697 521: 7,71 607: 11 1279: 72353441721527140856665601867 2203: 60449,1498429,711309659 2281: 197,557,1999,92033 3217: 11 4253: 7,53,8731,2353129,50820071 4423: 2163571 9689: 7,211,49922567 9941: 7,67,1605697,194147011 11213: 7 19937: 7,11,1129,168457 21701: 7 23209: 35603,620377 44497: 11,13259,16177141,896297147 86243: 7,29,301123,26072029 110503: 491,1493,1529761 132049: ? 216091: 4673,6920341 756839: 7 859433: 7 1257787: 11 1398269: 7,53,12713,17425081,199979189 2976221: 7,71 3021377: 7,11,49603 6972593: 7,6007,8392897,52193821 13466917: 11,45007 20996011: 1552147,114242767 24036583: 149 25964951: 7 30402457: 11 32582657: 7,11,67,34549,127541 So perfectnumber+1 are prime for p=2,3,13 and 19 and unknown for p=132049 (79502 digits) which I trialfactored to 1810⁹. Last fiddled with by ATH on 2008-05-28 at 11:08

2008-05-30, 00:30	#9
ATH Einyen Dec 2003 Denmark 6003₈ Posts	Question solved. I found a factor of 2^p-1(2^p-1) + 1 for p=132049 with gmp-ecm: 194528547122653 So of the 44 known perfect numbers P=2^p-1(2^p-1), P+1 is only prime for p=2,3,13 and 19. Last fiddled with by ATH on 2008-05-30 at 00:31

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2016-01-23, 09:38	#11
JeppeSN "Jeppe" Jan 2016 Denmark 10100110₂ Posts	Can we extend this? *p: factor(s) of 2p-1(2p-1) + 1** 57885161: 7 74207281: ? /JeppeSN