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Old 2022-07-05, 12:20   #1
Neptune
 
Jul 2022

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Default 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 formP_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} .
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Old 2022-07-05, 13:57   #2
mathwiz
 
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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
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Old 2022-07-05, 14:05   #3
ATH
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Quote:
Originally Posted by Neptune View Post
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.
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Old 2022-07-05, 14:10   #4
bur
 
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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.
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Old 2022-07-05, 14:36   #5
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Can “perfect number - 3” (except 3) be prime?
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Old 2022-07-05, 14:56   #6
charybdis
 
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Quote:
Originally Posted by sweety439 View Post
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.
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Old 2022-07-05, 20:46   #7
retina
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Quote:
Originally Posted by mathwiz View Post
Code:
... (2^60109187-1) ...
Is that a secret new prime that NSA/GCHQ has been hiding from us?
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Old 2022-07-05, 22:15   #8
kriesel
 
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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
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Old 2022-07-05, 23:43   #9
Batalov
 
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Quote:
Originally Posted by bur View Post
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!
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Old 2022-07-05, 23:56   #10
kriesel
 
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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
Click image for larger version

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Views:	9
Size:	6.8 KB
ID:	27080  

Last fiddled with by kriesel on 2022-07-06 at 00:29
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Old 2022-07-06, 00:11   #11
mathwiz
 
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Quote:
Originally Posted by retina View Post
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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