Problems & Puzzles: Puzzles

Puzzle 1095 RPP(74207281)

On July 13 2022, Andreas Höglund, sent two more results for our Puzzle 859.

As a matter of fact:

a) The Puzzle 859 is the third puzzle about the same issue, being the previous ones the puzzles 203 & 858.

b) The two results sent to my by Andreas, were obtained in 2008, and registered in the following thread: https://mersenneforum.org/showthread.php?t=10336

c) RPP(p)=(2^(p-1))*(2^p-1)+1, that is to say one unit more than the perfect number (2^(p-1))*(2^p-1) where (2^p-1) is the Mersenne prime number associated to the prime number p.

As is widely known, currently, there are only known 51 Mersenne primes.

In June 2002, Labos E found that RPP(p)=(2^(p-1))*(2^p-1)+1 is prime for p=2, 3 , 13 & 19 (7, 29, 33550337, 137438691329, respectively). All the other 47 cases have been found composite except RPP(74207281).

Up today, we don't know if RPP(74207281) it is prime or composite, by a primality test. And nobody has produced a factor for RPP(74207281).

Which means that there is only one hole in the carpet...

This is the complete status from the table published by Andreas Höglund:

p:		factor(s) of 2p-1*(2p-1) + 1
------------------------------------------------------------------
2:		prime
3:		prime
5:		7 , 71			(all factors)
7:		11 , 739		(all factors)
13:		prime
17:		7 , 11 , 111556741	(all factors)
19:		prime
31:		29 , 71 , 137 , 1621 , 5042777503				(all factors)
61:		2432582681 , 1092853292237112554142488617			(all factors)
89:		7 , 132599200423201647070231067 , 206381273143696885332153493	(all factors)
107:		7 , 11^2 , 67 , 231969487719072553476532687103891221170830716283757301763621	(all factors)
127:		11 , 107 , 261697, 70333627629913, 668114163064469436232560061443019245225783435495335057, (all factors)
521:		7 , 71, 1050252439763
607:		11, P365,(all factors)
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 , 2582471789
1398269:	7 , 53 , 12713 , 17425081 , 199979189
2976221:	7 , 71
3021377:	7 , 11 , 49603
6972593:	7 , 6007 , 8392897 , 52193821
13466917:	11 , 45007 , 6706083323
20996011:	1552147 , 114242767
24036583:	149
25964951:	7, 45850772753
30402457:	11 , 4654899979
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 < 2.3*10^13 and not known if it is prime or composite...
77232917:	7 , 11 , 11587
82589933:	7 , 67 , 599 , 7347113 , 14416229

Q. Can you try to prove that RPP(74207281) is prime or composite? If it is composite please send the least prime factor.


During the week 16-22 July, 2022, contributions came from Giorgio Kalogeropoulos.

***

Giorgio wrote:

I just found some more factors for smaller RPPs in case you want to update the table:

 
p:                 factor(s) of 2p-1*(2p-1) + 1
------------------------------------------------------------------
 
2203    ->      1659486433624153
2281    ->      22770375541
4253    ->      7883425250689, 19784944521553
4423    ->      1682337687991
9689    ->      880887102089
23209  ->     1285400531

***

 

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