mersenneforum.org Primes of the form 'perfect number' + 1 (a.k.a. RPN)
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 2022-07-05, 12:20 #1 Neptune   Jul 2022 112 Posts Primes of the form 'perfect number' + 1 (a.k.a. RPN) Let Pn denote the n-th even perfect number. Related to Mersenne primes, currently 51 perfect numbers are known. Primes of the form$P_n + 1$ are also listed in OEIS. Currently 4 primes are known: $P_1 + 1 = 2^1\cdot (2^2-1) +1 = 7$ $P_2 + 1 = 2^2\cdot (2^3-1) +1 = 29$ $P_5 + 1 = 2^{12}\cdot (2^{13}-1) +1 = 33550337$ $P_7 + 1 = 2^{18}\cdot (2^{19}-1) +1 = 137438691329$ Are all other cases up to $P_{51*}+1$ composite? Is a factor known of $P_{49*} + 1 = 2^{74207280}\cdot (2^{74207281}-1) +1$ ? I found no factor below $1.67 \cdot10^{12}$ .
 2022-07-05, 13:57 #2 mathwiz   Mar 2019 12316 Posts You can test this yourself with pfgw. Prepare a file like: Code: ABC 2^($a-1)*(2^$a-1)+1 521 607 1279 2203 2281 3217 4253 4423 9689 9941 11213 19937 21701 23209 44497 86243 110503 132049 216091 ... then: Code: ./pfgw64 -f10 -lpfgw.out -T8 ./nearperf.txt Output: Code: Recognized ABC Sieve file: 2^(521-1)*(2^521-1)+1 has factors: 7 2^(607-1)*(2^607-1)+1 has factors: 11 2^(1279-1)*(2^1279-1)+1 is composite: RES64: [570A6B3FD91E6339] (0.0154s+0.0010s) 2^(2203-1)*(2^2203-1)+1 is composite: RES64: [ECB4FE924C674723] (0.0244s+0.0010s) 2^(2281-1)*(2^2281-1)+1 has factors: 197 2^(3217-1)*(2^3217-1)+1 has factors: 11 2^(4253-1)*(2^4253-1)+1 has factors: 7 2^(4423-1)*(2^4423-1)+1 is composite: RES64: [F3603EEF4BD4F197] (0.0471s+0.0016s) 2^(9689-1)*(2^9689-1)+1 has factors: 7 2^(9941-1)*(2^9941-1)+1 has factors: 7 2^(11213-1)*(2^11213-1)+1 has factors: 7 2^(19937-1)*(2^19937-1)+1 has factors: 7 2^(21701-1)*(2^21701-1)+1 has factors: 7 2^(23209-1)*(2^23209-1)+1 has factors: 35603 2^(44497-1)*(2^44497-1)+1 has factors: 11 2^(86243-1)*(2^86243-1)+1 has factors: 7 2^(110503-1)*(2^110503-1)+1 has factors: 491 2^(132049-1)*(2^132049-1)+1 is composite: RES64: [1B3B60AEC3578817] (45.9414s+0.4677s) 2^(216091-1)*(2^216091-1)+1 has factors: 4673 2^(756839-1)*(2^756839-1)+1 has factors: 7 2^(859433-1)*(2^859433-1)+1 has factors: 7 2^(1257787-1)*(2^1257787-1)+1 has factors: 11 2^(1398269-1)*(2^1398269-1)+1 has factors: 7 2^(2976221-1)*(2^2976221-1)+1 has factors: 7 2^(3021377-1)*(2^3021377-1)+1 has factors: 7 2^(6972593-1)*(2^6972593-1)+1 has factors: 7 2^(13466917-1)*(2^13466917-1)+1 has factors: 11 2^(20996011-1)*(2^20996011-1)+1 has factors: 1552147 2^(24036583-1)*(2^24036583-1)+1 has factors: 149 2^(25964951-1)*(2^25964951-1)+1 has factors: 7 2^(30402457-1)*(2^30402457-1)+1 has factors: 11^2 2^(32582657-1)*(2^32582657-1)+1 has factors: 7 2^(37156667-1)*(2^37156667-1)+1 has factors: 7 2^(42643801-1)*(2^42643801-1)+1 has factors: 3593 2^(43112609-1)*(2^43112609-1)+1 has factors: 7 2^(57885161-1)*(2^57885161-1)+1 has factors: 7 2^(60109187-1)*(2^60109187-1)+1 has factors: 7
2022-07-05, 14:05   #3
ATH
Einyen

Dec 2003
Denmark

3,347 Posts

Quote:
 Originally Posted by Neptune Is a factor known of $P_{49*} + 1 = 2^{74207280}\cdot (2^{74207281}-1) +1$ ? I found no factor below $1.67 \cdot10^{12}$ .
We have an old thread for this:

The list is in post #10, we are only missing factor for that one for p=74207281. I checked to 6*1012 back then when that Mersenne Prime was found.

 2022-07-05, 14:10 #4 bur     Aug 2020 79*6581e-4;3*2539e-3 52·23 Posts These should have very low weight due to k being a very large prime. Did you test for factors or have factors of 7 < n < 49? LLR2 didn't like even the 8-th one: Code: \$ ./sllr2 -d -q"2147483647*2^30+1" 2147483647 > 2^30, so we can only do a PRP test for 2147483647*2^30+1. Iter: 1/31, ERROR: ILLEGAL SUMOUT Possible hardware failure, consult the readme file. Continuing from last save file. Waiting five minutes before restarting.
 2022-07-05, 14:36 #5 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 3,559 Posts Can “perfect number - 3” (except 3) be prime?
2022-07-05, 14:56   #6
charybdis

Apr 2020

79310 Posts

Quote:
 Originally Posted by sweety439 Can “perfect number - 3” (except 3) be prime?
No.

2p-1(2p-1)-3 = (2p-1+1)(2p-3)

And if an odd perfect number exists then subtracting 3 gives an even number.

2022-07-05, 20:46   #7
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

655810 Posts

Quote:
 Originally Posted by mathwiz Code: ... (2^60109187-1) ...
Is that a secret new prime that NSA/GCHQ has been hiding from us?

 2022-07-05, 22:15 #8 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 22·1,667 Posts First LL by "b3g" https://www.mersenne.org/report_expo...exp_hi=&full=1 which stands for "beyond third generation" Last fiddled with by kriesel on 2022-07-05 at 22:19
2022-07-05, 23:43   #9
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

990110 Posts

Quote:
 Originally Posted by bur These should have very low weight due to k being a very large prime.
This is, as some folks say, "not even wrong".

/sigh/ There is so much excitement doing something that was done to death years ago, right?

 2022-07-05, 23:56 #10 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 22·1,667 Posts A current link, as a substitute for the dead link for the sequence in https://www.mersenneforum.org/showpo...61&postcount=3 is https://oeis.org/search?q=a061644&la...lish&go=Search or more concisely, https://oeis.org/A061644 edit: which I see has now been updated in place, silently (without note by whichever mod edited that 14 year old post this evening). Old link shown with resulting error response below. Apparently a valid link when originally posted, from before the transition from AT&T to OEIS.org in 2009-2010. https://oeis.org/wiki/Welcome#OEIS:_Brief_History Attached Thumbnails   Last fiddled with by kriesel on 2022-07-06 at 00:29
2022-07-06, 00:11   #11
mathwiz

Mar 2019

3×97 Posts

Quote:
 Originally Posted by retina Is that a secret new prime that NSA/GCHQ has been hiding from us?
Ha. That's because I foolishly grabbed this list instead of going to mersenne.org, and fed it through sed + awk without checking what I was doing.

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