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.

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

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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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In Puzzle 859 I wrote:

On January 4, 2023, Martin Hopf wrote:

Q3: Can you try to prove that RPP(74207281) is prime or composite? ...

 

It is a proven composite!

On December 23rd 2022, a 17-digit factor was found with the elliptic curve method:
 

14344999215792989

Sorry for this last update on this page (Nov 8, 2024

BTW, Officially since Oct 12 2024, the newest and largest Mersenne prime is M(136279841). See https://www.mersenne.org/primes/?press=M136279841.

Accordingly RPP(136279841) could be prime or composite. See Puzzle 203.

Applying the Wilke rules, See Puzzle 203, RPP(136279841) is composite divided by 7 because mod(6*136279841-5,7)=0.

 

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