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 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, 14:36   #2
R.D. Silverman

Nov 2003

22·5·373 Posts

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.

2008-05-27, 15:45   #3
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

101100101012 Posts

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

 2008-05-27, 16:07 #4 petrw1 1976 Toyota Corona years forever!     "Wayne" Nov 2006 Saskatchewan, Canada 5×907 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 300910 Posts I trialfactored P+1 for the 44 known perfect numbers P and did ECM on 1 of them: p: factor(s) of 2p-1*(2p-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. Last fiddled with by ATH on 2008-05-28 at 11:08
2008-05-28, 11:22   #6
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

101100101012 Posts

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.

Last fiddled with by R. Gerbicz on 2008-05-28 at 11:23

2008-05-29, 08:25   #7
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

2×2,897 Posts

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 2p-1*(2p-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

2008-05-29, 16:59   #8
philmoore

"Phil"
Sep 2002
Tracktown, U.S.A.

2·13·43 Posts

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.

 2008-05-30, 00:30 #9 ATH Einyen     Dec 2003 Denmark 3·17·59 Posts Question solved. I found a factor of 2p-1*(2p-1) + 1 for p=132049 with gmp-ecm: 194528547122653 So of the 44 known perfect numbers P=2p-1*(2p-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
 2008-09-16, 23:19 #10 ATH Einyen     Dec 2003 Denmark 3·17·59 Posts Updated list: Code: p: factor(s) of 2p-1*(2p-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=2p-1*(2p-1), P+1 is only still prime for p=2,3,13 and 19, and status unknown for p=74207281. Last fiddled with by ATH on 2020-01-30 at 03:13
 2016-01-23, 09:38 #11 JeppeSN     "Jeppe" Jan 2016 Denmark 22·41 Posts Can we extend this? p: factor(s) of 2p-1*(2p-1) + 1 57885161: 7 74207281: ? /JeppeSN

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