Mersenne Semiprimes

[Batalov]

Well, no surprises here, it's composite:

M125731369/"1" interim We8 residue EE9CB5973F87EBC5 at iteration 123000000
M125731369/"1" interim We8 residue ED1BBFF1E242A1AC at iteration 124000000
M125731369/"1" interim We8 residue FF77A5DC9D544CBD at iteration 125000000
[Wed Feb 10 22:50:05 2016]
M125731369/M11213 is not prime.  RES64: 7B39D9067F3A3A06. We8: 00000000,00000000

[ATH]

Good to know for sure. Nice work making it possible to test them in a reasonable amount of time.

[Prime95]

Originally Posted by Batalov

EDIT: this patch-2.0 works much better:

Attached is the updated ecm.c file. It includes this upgrade and another of your Phi changes.

[ATH]

Sieved M(77232917^2) / M(77232917) without finding a factor and sieved some of the lower ones some more without any new factors..

p		Factor(s) of M(p^2)/M(p)			k in 2*k*p^2+1
----------------------------------------------------------------------------------------
2		Prime
3		Prime
5		601,1801					k=12,36
7		Prime
13		4057						k=12
17		12761663					k=22079
19		9522401530937					k=13188921788
31		280651416271709745866686729			k=146020507945738681512324
61		80730817,301780543,281646330073			k=10848,40551,37845516
89		29123869433,49849688719				k=1838396,3146679
107		1167799,377175857				k=51,16472
127		14806423,25044595073,72653532113		k=459,776384,2252264
521		8143231,10857641,4338170063			k=15,20,7991
607		345899921201,166969148315503			k=469400,226583799
1279		103097448872275370551				k=31512062869275
2203		15714690743					k=1619
2281		Composite	(No factor 2*k*p^2+1 < 2^73)	(k<90*10^13))
3217		102559471991					k=4955
4253		1844976919,57592220657				k=51,1592
4423		Composite	(No factor 2*k*p^2+1 < 2^74)	(k<48*10^13))
9689		76729816024661281759				k=408673285599
9941		Composite	(No factor 2*k*p^2+1 < 2^76)	(k<38*10^13))
11213		Composite	(No factor 2*k*p^2+1 < 2^77)	(k<60*10^13))
19937		Unknown		(No factor 2*k*p^2+1 < 2^78)	(k<38*10^13))
21701		33907204873,153745627424471			k=36,163235
23209		17206738756236217				k=15971868
44497		Unknown		(No factor 2*k*p^2+1 < 2^81)	(k<61*10^13))
86243		Unknown		(No factor 2*k*p^2+1 < 2^77.9)	(k<2*10^13))
110503		250836575030879,22513968547647823		k=10271,921879
132049		Unknown		(No factor 2*k*p^2+1 < 2^79.2)	(k<2*10^13))
216091		Unknown		(No factor 2*k*p^2+1 < 2^80.6)	(k<2*10^13))
756839		696531210655937,63659341689518360417		k=1216,55568048
859433		17727001955737,667717073666057			k=12,452
1257787		Unknown		(No factor 2*k*p^2+1 < 2^85.7)	(k<2*10^13))
1398269		34207412811532057				k=8748
2976221		Unknown		(No factor 2*k*p^2+1 < 2^88.1)	(k<2*10^13))
3021377		2329356963700884673				k=127584
6972593		Unknown		(No factor 2*k*p^2+1 < 2^90.6)	(k<2*10^13))
13466917	Unknown		(No factor 2*k*p^2+1 < 2^92.5)	(k<2*10^13))
20996011	899298254940726841				k=1020
24036583	5274651651287933470393				k=4564764
25964951	20225360412972031				k=15
30402457	Unknown		(No factor 2*k*p^2+1 < 2^94.9)	(k<2*10^13))
32582657	3322246487577398706217				k=1564692
37156667	71776963464264825905447				k=25994507
42643801	901972906808890097				k=248
43112609	Unknown		(No factor 2*k*p^2+1 < 2^95.9)	(k<2*10^13))
57885161	Unknown		(No factor 2*k*p^2+1 < 2^96.7)	(k<2*10^13))
74207281	508014103943653104553301983			k=46126737231
77232917	Unknown		(No factor 2*k*p^2+1 < 2^97.6)	(k<2*10^13))

[LaurV]

the small ones (13, 17, 19) can be easily full factored (19^2-19=384-19 bits, about 100 digits)

gp > factorint(m2(p))
time = 647 ms.
[                           4057 1]
[                  6740339310641 1]
[3340762283952395329506327023033 1]

[GP2]

Originally Posted by Jens K Andersen

Getting back to the pointless computations instead of discussing their pointlessness, at http://donovanjohnson.com/mersenne.html I think I found the 5 largest known probable Mersenne semiprimes :
M(684127) = 23765203727 * prp205933
M(406583) = 813167 * prp122388
M(271549) = 238749682487 * prp81734
M(271211) = 613961495159 * prp81631
M(221509) = 292391881 * prp66673

The largest proven Mersenne semiprime at http://primes.utm.edu/top20/page.php?id=49 is:
M(17029) = 418879343 * p5118

Updating the above post from 2010, which is about the boring kind:

The original list above is missing the entries:

M(611999) = 18464214225958267477777390354183 * prp184199
M(432457) = 1672739247834685086279697 * prp130159

Of course, every unfactored Mersenne number of prime exponent is potentially a semiprime, so more entries may be added at any time as new factors are found.


The largest known probable Mersenne semiprimes are:

M(7313983) = 305492080276193 * prp2201714
M(5240707) = 75392810903 * prp1577600
M(4187251) = 72234342371519 * prp1260475
M(3464473) = 604874508299177 * prp1042896
M(2327417) = 23915387348002001 * prp700606

The largest proven Mersenne semiprime is:

M(63703) = 42808417 * p19169

and the next semi-feasible candidate for Primo certification is:

M(86371) = 41681512921035887 * prp25984

[alpertron]

Originally Posted by GP2

Updating the above post from 2010, which is about the boring kind:

The original list above is missing the entries:

M(611999) = 18464214225958267477777390354183 * prp184199
M(432457) = 1672739247834685086279697 * prp130159

These do not appear in that list because they were found after 2010. There are lots of Mersenne semiprimes, but the problem is to find the smallest prime factor.

PS. Maybe in the future someone will find an algorithm to quickly detect probable semiprimes. In http://physics.open.ac.uk/~dbroadhu/cert/semgpch.gp you can find a 5061-digit proven semiprime, but the certificate does not need the prime factors.