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?

[QUOTE=Housemouse;134548]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?[/QUOTE]

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

[QUOTE=Housemouse;134548]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?[/QUOTE]

[URL="http://www.research.att.com/~njas/sequences/A061644"]http://www.research.att.com/~njas/sequences/A061644[/URL]

yes
 

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

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

That is as far as I checked

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

[U]p: factor(s) of 2[sup]p-1[/sup]*(2[sup]p[/sup]-1) + 1[/U]
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*10[sup]9[/sup].

[QUOTE=ATH;134626]
So perfectnumber+1 are prime for p=2,3,13 and 19 and unknown for p=132049 (79502 digits) which I trialfactored to 18*10[sup]9[/sup].[/QUOTE]

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.

[quote=ATH;134626]I trialfactored P+1 for the 44 known perfect numbers P and did ECM on 1 of them:

[U]p: factor(s) of 2[sup]p-1[/sup]*(2[sup]p[/sup]-1) + 1[/U]
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*10[sup]9[/sup].[/quote]
gmp-ecm doesnt think p=132049 is prp

[QUOTE=henryzz;134683]gmp-ecm doesnt think p=132049 is prp[/QUOTE]

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

Question solved.

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

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

Updated list:

[CODE][B]p: factor(s) of 2[sup]p-1[/sup]*(2[sup]p[/sup]-1) + 1[/B]
------------------------------------------------------------------
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*10[sup]12[/sup]
77232917: 7 , 11 , 11587
82589933: 7 , 67 , 599 , 7347113 , 14416229
[/CODE]

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

Can we extend this?

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

/JeppeSN