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 3 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 4478 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

23×419 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:
https://www.mersenneforum.org/showthread.php?t=10336

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 2×293 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 72×73 Posts Can “perfect number - 3” (except 3) be prime?
2022-07-05, 14:56   #6
charybdis

Apr 2020

19×43 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

2·17·193 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 22×1,669 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

22·2,477 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?
Much more excitement than to read the link that was helpfully provided. That thread has everything you ever wanted to know. Read it!

 2022-07-05, 23:56 #10 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 22·1,669 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

5×59 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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