Prime95 - suggest using B2 bound = GMP-ECM default and other questions

[Raman]

Prime95 - suggest using B2 bound = GMP-ECM default and other opinions

Since Monday 15 February 2016 I have been using Prime95 to extract ECM factors from large Mersenne composite numbers, and I have found out 75 so far (and counting)...
Honestly, this ECM on small Mersenne composite numbers is the only work that could probably produce useful results with little amount of effort is being needed without wasting out any computing power...
ECM on numbers like 2¹²⁷⁷-1, 2¹⁶¹⁹-1, 2¹⁷⁵³-1, 2²¹³⁷-1, 2²²⁶⁷-1, 2²²⁷³-1, 2²³⁵⁷-1, 2²³⁷⁷-1, 2²⁴²³-1, 2²⁴⁷⁷-1, 2²⁵²¹-1, 2²⁵⁵⁷-1, 2²⁶⁷¹-1, 2²⁷¹³-1, 2²⁷¹⁹-1, 2²⁸⁵¹-1, 2³⁰⁴⁹-1, etc.
will probably be totally futile without producing any useful results.
On the other hand, the regular Cunningham tables have been thoroughly ran away / executed away with the ECM curves by using other people - be being!
Any way completely factoring the Mersenne composite numbers that I am running ECM curves on will take ages
unless integer factorization is in P or advances in computing power have been made or quantum computers have been developed and implemented or advances in integer factorization algorithms have been made.

Here are the 75 (and counting) factors that I have extracted from large Mersenne composite numbers.

1. 2¹⁵⁸²⁰⁹-1 has a factor: 26496805856040782582748460014520511
2. 2^1413017-1 has a factor: 2226694532490185824727
3. 2^1320091-1 has a factor: 231827487452450337577
4. 2^1085809-1 has a factor: 8381291184102382541497
5. 2^1420109-1 has a factor: 85148369868485192804575096841
6. 2^1306477-1 has a factor: 2886593156080095874622681
7. 2^1264387-1 has a factor: 111421197216700721651761
8. 2^1073647-1 has a factor: 45496627768358111930287
9. 2^1297129-1 has a factor: 21635389012955382990854233
10. 2^1249741-1 has a factor: 168691067410297172399521
11. 2^1415957-1 has a factor: 49084592298749387589959
12. 2^1205173-1 has a factor: 269867708244063132649
13. 2^1335563-1 has a factor: 1059938762839012963385177
14. 2^1110167-1 has a factor: 14693675794413895601977
15. 2^1185697-1 has a factor: 103519616241334574089
16. 2^1185319-1 has a factor: 187769929174579865730719
17. 2⁶⁸⁰⁰⁷⁷-1 has a factor: 359004574534541931650318449 (σ = 1545834309296023)
18. 2^1406857-1 has a factor: 76972723442421389411833
19. 2^1103281-1 has a factor: 461658571972334163101951
20. 2^1082533-1 has a factor: 1623069162917486699809
21. 2^1001629-1 has a factor: 25980809010119884941817
22. 2⁶⁷⁹⁶⁶⁹-1 has a factor: 63260117764832948321737 (σ = 5074446274686185)
23. 2^1409633-1 has a factor: 4562270691505620280350953
24. 2^1314161-1 has a factor: 200209995939544733759
25. 2^1401767-1 has a factor: 208164200904253362898039
26. 2^1309811-1 has a factor: 6452094516571781903870908529
27. 2^1212769-1 has a factor: 22686603584990095872353327
28. 2^1155919-1 has a factor: 2446997794177577666902847
29. 2⁶⁷⁴⁴⁸³-1 has a factor: 1185762874913395945847 (σ = 8472293866228076)
30. 2^1151041-1 has a factor: 98338208354439540871
31. 2^1389149-1 has a factor: 1968838610224783811144921
32. 2^1362349-1 has a factor: 4052494766327711626877807
33. 2⁶⁴⁰⁴⁸³-1 has a factor: 35948758457297251395594732823879 (σ = 7385969443002972)
34. 2^1238119-1 has a factor: 19398358097550480988759
35. 2^1361609-1 has a factor: 8394136703834908414711
36. 2^1381381-1 has a factor: 486078017602417074080489
37. 2¹³⁰⁵⁴⁷-1 has a factor: 740710078242573288550675295986147001
38. 2^1352543-1 has a factor: 54914730373539406420367
39. 2^1048361-1 has a factor: 317587687414956893560559
40. 2^1062367-1 has a factor: 5715626615941288867913951
41. 2^1209781-1 has a factor: 61574197954768721757880409
42. 2⁶¹⁴⁹⁰⁹-1 has a factor: 2157432123233208899520343 (σ = 5192865221372438)
43. 2^1017043-1 has a factor: 2714087245272479000359
44. 2^1403807-1 has a factor: 910040398878981948860243369
45. 2^1151327-1 has a factor: 374909084903502844569223
46. 2^1394089-1 has a factor: 12522193154648380174007
47. 2^1049891-1 has a factor: 14792762552601957907766311
48. 2⁶¹⁹⁹⁷⁹-1 has a factor: 39952445419572244259873 (σ = 8735607385635865)
49. 2^1244249-1 has a factor: 440414964773453095807
50. 2^1176221-1 has a factor: 461610300972747494021593
51. 2^1030219-1 has a factor: 3772589184204120078374871601
52. 2^1226387-1 has a factor: 1074355317155260325278943
53. 2^1062793-1 has a factor: 12513099136956335921312007791
54. 2^1023769-1 has a factor: 121894953460427063046553
55. 2^1196059-1 has a factor: 17663439686039008542085063 (σ = 1539397084932662)
56. 2^1111949-1 has a factor: 131359229602139901573898435183
57. 2^1328891-1 has a factor: 190486443247254756345034440671
58. 2^1198261-1 has a factor: 186410816493618076522807 (σ = 933907529783551)
59. 2^1058077-1 has a factor: 1155399346576072505554663
60. 2^1295611-1 has a factor: 1538613486295112984311
61. 2^1387681-1 has a factor: 7635519794781065554162223 (σ = 7915873887005268)
62. 2^1293421-1 has a factor: 46804080444409785497993
63. 2^1094623-1 has a factor: 7153958318333686172617
64. 2^1409549-1 has a factor: 6884935783696551152303
65. 2^1045487-1 has a factor: 12870043214611775340199
66. 2^1196123-1 has a factor: 25412328569397020505047
67. 2^1104017-1 has a factor: 10390283462108941247777
68. 2^1396529-1 has a factor: 3430327700879658519722567
69. 2^1269167-1 has a factor: 69389537933503942909169
70. 2^1357901-1 has a factor: 4006306700470164867761
71. 2^1177741-1 has a factor: 1393309518989601569849
72. 2^1172207-1 has a factor: 3612667936408840580839
73. 2⁶⁸³⁸³¹-1 has a factor: 817118841184883531740073 (σ = 1237148984876076)
74. 2^1342907-1 has a factor: 3583767036379540468879
75. 2⁶⁹²⁷⁷⁹-1 has a factor: 9113973384791662609433 (σ = 8220512900751494)

So, roughly 280 ECM curves per factor at 25 digit level with B1 = 50000 and B2 = 5000000
and roughly 4700 ECM curves per factor at 35 digit level with B1 = 1000000 and B2 = 100000000.
Does PrimeNet server assign exponents range for Prime95 for ECM curves of small Mersenne composite numbers based up on the amount of memory allocated?
I initially allocated most of the machines with memory = 128 MB, some with 256 MB, a few with 512 MB and 1024 MB.
My own system with memory = 64 MB.
But, one of the machines allocated with 1024 MB had reset to 8 MB and later it found out its own range level of factors of
2¹³⁰⁵⁴⁷-1 has a factor: 740710078242573288550675295986147001 and then
2¹⁵⁸²⁰⁹-1 has a factor: 26496805856040782582748460014520511 or some 4772 ECM curves later.

By the way, why did Prime Net Server once assign one of single systems and then computers or machines for Trial Factoring assignment
when I was looking out only for ECM curves on to smaller Mersenne composite numbers assignment?

[Raman]

I would like to give a few of my suggestions to make GIMPS Project better.

1. At the first place, I would suggest that Prime95 would use a B2 bound of GMP-ECM default and not 100 × B1.
   Though it is not of much of difference for smaller factors, but as the factors get bigger, this would be a serious bottleneck.
   For small Fermat numbers and Mersenne numbers without any known factors or are incompletely factored, see the difference between Prime95 and GMP-ECM.
   For example, consider at the 65 digit level, with B1 = 850000000, GMP-ECM requires only 69471 ECM curves while Prime95 requires over 360000 ECM curves.
   What a waste of computing power!
   Who knows it that how many unknown factors of Fermat numbers and Mersenne numbers did we miss out with Prime95 because of this serious issue that which are without any known factors or are incompletely factored!
   Please fix this serious issue immediately, as soon as possible!
   Manually, I can set up with in the worktodo.txt file to set up with B2 bound = GMP-ECM default, but when running automatically over PrimeNet server over multiple systems, this becomes a major problem!
   How many people without this knowledge are running with Prime95 out side with B2 is being set to 100 × B1 range levels!
   May be that let this thing not happen at all with in future times only certainly which it is being a serious issue and then, or a major problem that way!
   For example, please consider with in:
   2⁶⁴⁰⁴⁸³-1 has a factor: 35948758457297251395594732823879 (σ = 7385969443002972).
   Its own group order is being: 2² × 3⁴ × 29 × 167 × 617 × 65173 × 79229 × 130253 × 55207697.
   At the 30 digit level, with in B1 = 250000 and B2 = 25000000 computing range level, we would only certainly miss out with in getting seeking this factor.
   Had been B2 = GMP-ECM default range level, I am being able to get catch it that factor tight that which would not be let out loose only certainly.
   Should consider running with Prime95 for stage 1 and GMP-ECM for stage 2 for getting with maximum through put!
2. Secondly, I would suggest that PrimeNet server assigns tasks with ECM on small Mersenne numbers with not only on Mersenne composite numbers with no known factors but also which are partially factored.
   Partially factored Mersenne composite numbers have higher probability of extracting factor by using ECM curves, because fewer people might have touched up on these candidates.
   Nevertheless, I had observed that TJAOI has been working out actively up on numbers of these types.
   Who is TJAOI? Is he a member of this forum? It would be good to have a talk and discussion with him in side!
   It would be good if mersenne.org web site page records prime factors of composite numbers of form 2^p-1 where p is not necessarily being prime number at all!
3. 1. Thirdly, for Wagstaff numbers like (2⁸¹⁹¹+1)/3, (2¹⁹⁹³⁷+1)/3, (2¹¹⁰⁵⁰³+1)/3, (2⁵²⁴²⁸⁷+1)/3
      Prime95 just outputs the factor of 3 and then stops, or while I am looking out for the non-trivial factors.
      It does not parse away the factor of 3 properly mentioned away properly in the worktodo.txt file in the ECM2 line finally - (not in the ECM line at all) at all!
      It is indeed possible to turn off this option with in Continue ECM flag only certainly.
      But, the first ECM curve is some times being always getting wasted away.
      This effect is being especially enhanced if we need to restart with in the software application frequently often.
      One single ECM curve rather takes away with in a significant amount of time only certainly with higher values of B1 and B2 bounds - range level limits - rate scale ratio proportion so that,
      such that this issue becomes further more significant!
   2. How about giving away with in fractional values of k for composite numbers of form k×(b^p±ε) only certainly?
4. Is Prime95 efficient than GMP-ECM in stage 1 and GMP-ECM efficient than Prime95 in stage 2?
   By using GMP-ECM Hook flag, how much efficient is Prime95 than GMP-ECM in stage 1 and how much efficient is GMP-ECM than Prime95 in stage 2?
   for factors of all sizes and input numbers of all sizes?
5. Other less important, minor issues and other different questions:
   1. Prime95 could use the prefix of W for Wagstaff numbers, as with M for Mersenne numbers. Older versions of Prime95 used P instead of W, any way that newer version use the syntax of 2^p+1, how ever that trivial factor of 3 can be pumped as long as exponent p is being odd!
   2. In Prime95 p-1 and ECM menu, why does the exponent still remain as 1061? 2¹⁰⁶¹-1 has already been completely factored over 4 years ago by now, and it could be changed to some thing like 1277. How ever I believe that factoring 2¹²⁷⁷-1 is unlikely to produce any productive results, even when B2 has been set to GMP-ECM default range level.
   3. In Prime95 Time menu, input exponent is given as 38000000 and in Test Menu, input exponent is given as 0. Could they be set to valid prime numbers?
   4. What is the difference between ECM and ECM2 in to the worktodo.txt file?
   5. Consider readme.txt file:
      Factoring M400037 to 2^54 is 3.02% complete. Time: 0.121 sec.
      First of all the exponent of 400037 is not being prime at all. What is the point in stupidly testing away the primality of this number by using factoring it? Could they be set to valid prime numbers?
      That which of exponent must be a prime between 5 and 560000000, the upper limit is now being 1000000000.
      Exponent of 876543 is being mentioned away with in of in side of readme.txt file which again that it is being not a prime number at all that way!
   6. Why does ECM Progress web site page still contain exponents like 1409 which are being completely factored away right now?
   7. Why does some stupid guys run ECM curves, p-1 and p+1 on numbers like 2¹⁰⁶¹-1 even after which it had been completely factored away?
   8. Why does some stupid guys run ECM curves, p-1 and p+1 on prime number candidates like 2¹¹²¹³-1, 2¹⁹⁹³⁷-1, 2²¹⁷⁰¹-1, 2²³²⁰⁹-1, 2⁴⁴⁴⁹⁷-1, 2⁸⁶²⁴³-1, etc.?
   9. Does any factorization log file give away first larger factor below smaller factor?
   10. Samuel Wagstaff's Cunningham table home page contains 10 most wanted numbers and 24 more wanted numbers. What is being the secret behind this selection of the numbers 10 and 24?
   11. Or then in Prime95 in the ECM2 line - (not in the ECM line at all) - at all - it accepts the known factors of a composite number of the form k×(b^p±ε) finally, but the larger factors get rounded off. Why? Like the factors of 513668017883326358119 and 883852340565787164089923172087 for 2⁶⁵⁵³⁷-1.
   12. Does Prime95 accept a B1 and B2 bound of greater than 4294967296? In the earlier days, other people would use a B1 and B2 bound of lesser than this value, which that it is being even further more bad, yet, still, that way!
   13. Prime95 in Windows has got a Boot On Start Up option. Does mprime in Linux or Unix have got the same option?
   14. Could mersenne.org web site page in the future times record with in the σ value for every ECM curve ran away or executed away whose successful factor is being found out some times by using the other people only certainly up right now?
6. Once I had observed that in the Factors Found page, TJAOI was able to find out a continuous sequence of factors in strictly increasing order that divide some Mersenne composite number 2^p-1 for some prime number p ≤ 10⁹.
   Does any one over here by has got an idea of what algorithm or script is he being running away or executing away?
7. 1. Can there exist a small prime factor whose group order is being bad, i.e. not smooth for any chosen value of σ?
   2. Is it possible that for a large prime factor, the group order is being the same for two different chosen values of σ? Or group order = p-1? Or group order = p+1? If so, then what is being the probability or likelihood?
   3. Can we find out a clever value of σ before hand such that the group order is being smooth for some unknown value of prime factor? How does ECM curves in Prime95 and GMP-ECM choose away a value of σ? Randomly?
   4. Why does Prime95 use a 16 digit value of σ, where as GMP-ECM (older version) uses a 10 digit value of σ usually ≤ 4294967296. Why? Newer version of GMP-ECM, once I had observed that generates a value of σ such that as some ratio? So that, what is being that thing?
   5. Does ECM fail to find out a prime factor if all the prime factors of group order are below B2 but two or more of the prime factors of group order are above B1?
   6. Does ECM fail to find out any of the two prime factors if their own product is being (B1, B2) - smooth, but the two individual prime factors by themselves are being not necessarily (B1, B2) - smooth?
   7. Can some one provide me with a PARI/GP script for calculating ECM curve group order from a some prime factor and a some value of σ?
      We have got Magma script from Paul Zimmermann's GMP-ECM curves web site page - but that Magma is being a paid package - for obtaining full version!
      Factordb.com web site page accepts calculating ECM curves group order from a some prime factor and a some value of σ up to 100 digits right now, formerly up to 90 digits.
8. 1. If p and q are two different prime numbers so that, such that both of their own largest prime factors of p-1 and q-1 are being exactly the same essentially. p-1 will fail away because of degenerate case. Will p+1 find out the individual factors with in this case a lot easily only certainly? Why? What about ECM curves? With in some difficulty range level only certainly?
   2. For Cunningham tables factoring, one of my suggestions for ECM curves is to give away the input number as product of all candidate holes. This product has plenty of factors and then, or smaller factors will hit frequently often for ECM curves for a some value of σ or other.
      And then, for Cunningham tables factoring, candidate holes 2,1213-, 2,1213+, 2,2426L, 2,2426M
      yet you could still use special division for up on a some 16¹²¹³-1 and then or extract smaller factors from one of these 4 numbers simultaneously for a some value of σ. Have other people got this idea already seeked - obtained / up looked out - attained up - attained?
   3. It would be good if a some of the larger Mersenne composite number candidates get some p+1 and ECM curves ran away / executed away after wards of trial factoring and then p-1, or before sending them in to a first time Lucas Lehmer trial test run away / executed away.
   4. It would be good if mersenne.org web site page keeps away a some factor repository of the Wagstaff composite number candidates prime factors, and then or other bases b also too!
      Factoring such Cunningham numbers are always being useful.
      For prime number bases b, factoring away composite numbers of the following form b^p-1 reveals away with group structure of GF(b^p) whose group order of any element of that group will always be a divisor of b^p-1, whether p is being a prime number or p is being a composite number. Why? And then what are being the other applications? Think about it!
      b^p+1 is always being a factor of b^2p-1, just!
   5. For base b, expect a some, one single prime number candidate of the following form (bⁿ-1)/(b-1) and then or (bⁿ+1)/(b+1) between n = x and n = e^γx, for some value of x, where by over here by γ is being standing away for the Euler-Mascheroni constant - where by over here by!
9. In Prime95, why does floating point numbers are being used for the Lucas Lehmer trial test that they are being run away / executed away? Why?
   Rather than, instead of pure integers? What is being the purpose of double checking?
   What is being the purpose of rotating away / shifting away bits initially?
   Why it is being extremely unlikely for two incorrectly running away / executing away Lucas Lehmer trial tests up to produce same residues that they are being this case? Why?
10. Last but not the least, what changes were made up on to GMP-ECM version 7.0?
    Why does it take away with in a larger number of ECM curves for the same N-digit level at the same B1 bound than when compared against GMP-ECM version 6.0 - only certainly - last but not the least?

[Raman]

1. During the 20 year tenure period of the GIMPS Project, has it run across into any Lucas Lehmer test false positives and composite number exponents with residue (usually last 64 bits) 0 or any Lucas Lehmer test false negatives and prime number exponents with residue (usually last 64 bits) not 0? The former three cases should have been possible while the latter fourth cases should not have been possible.
2. Can I have a Mersenne Forum badge for latest prime number found out 2^74207281-1? I know that it has been over 9 months, but if it were being still available, as yet, my new address - house is being shifted out away - up:
   Raman Viswanathan,
   Flat Number 13D, Plot Number 44,
   North Park Street,
   Elim Nagar, Perungudi,
   Chennai - 600 096.
   Tamiɻ Nadu, India.
   12.9618° North, 80.2382° East.
   A some mersenne forum T-shirt that is being containing with in (front side) Mersenne prime numbers (back side) Wagstaff prime numbers (right hand side sleeve) Repunit prime numbers (left hand side sleeve) Prime numbers p for which (10^p+1)/11 is being a prime number only certainly that would be indeed obviously very much all most good and then or productive enough rather than instead of a some mersenne forum badge - process and then or phenomenon!
   Glow T-shirts are also being - will too be for good and interesting!
   Besides of similarly on to glow paints are also being - will too be for painted of over ceiling!
3. How did Joppe Bos, et. al. team able to finish off 2,1193- and all other Cunningham 2- tables extremely difficult numbers while I was being away from this mersenne forum? By using Coppersmith SNFS. They used simultaneous sieving with a same SNFS algebraic polynomial shared. Why did they only attack the Cunningham 2- tables, and not the other bases, 2+ and 2LM? How much resources do they have with? Or that their own method is being workable out away only up for base 2 numbers or all other bases? Any way that 2,(p)+ is being a factor of 2,(2p)- and that 2,(2p)L and 2,(2p)M is being a factor of 2,(4p)-. At their own will, they could finish off with in the entire Cunningham tables, Fibonacci and Lucas numbers, Homogeneous Cunningham numbers and other twisted additive and multiplicative groups like these type of things. What are they doing with right now? May be that in the future times, NFS@Home could complete off entire Cunningham tables by using Coppersmith SNFS, be being with ≥ 50% savings of computing power! But, for smaller numbers as those NFS@Home is being working out up on right now, is Coppersmith SNFS being sub optimal?
4. Does the following fraction of ¾ have got with an official proper name? If not so, then the name triquarter would be very much fancier all though, of even though.
5. Why is oddperfect.org web site page being of still running ECM curves upon (13²⁶⁹-1)/12, as of yet? It is being right now of completely factored. Right?
6. Will quantum computers eventually become a reality? When? If Shor's algorithm uses with Quantum Fourier Transform to find out the period Φ(N) over a quantum computer, then why cannot we use with Classical Fourier Transform to find out the period Φ(N) up on a classical computer?
7. Consider with Mersenne Numbers of the following form 2^p-1, there are being plenty of prime numbers of the following form 2kp+1 but only a few of them would divide 2^p-1 such that their own product ≤ 2^p-1. Due to a theorem of Number Theory, a prime number of the following form 2kp+1 should indeed obviously divide with 2^kp-1. So that, 2kp+1 should divide with 2^p-e^nπi/k for a some integer n such that 0 ≤ n ≤ 2k-1. For example, 367 = (2 × 3 × 61) + 1 neither divides with 2⁶¹-1 nor divides with 2⁶¹+1 but it should divide with 2³⁶⁶-1 = (2⁶¹-1) × (2⁶¹+1) × (2¹²²-2⁶¹+1) × (2¹²²+2⁶¹+1). Indeed 367 divides with (2¹²²+2⁶¹+1).
   For k = 1 case, this reduces in to if 2p+1 is being a some prime number for a some prime number p, then it should divide with either 2^p-1 (when 2 is being quadratic residue (mod 2p+1), 2p+1 ≡ 1 or 7 (mod 8)), or divide with 2^p+1 (when 2 is being quadratic non-residue (mod 2p+1), 2p+1 ≡ 3 or 5 (mod 8)). (p that is being Sophie-Germain prime number. 2p+1 is also being prime number too!)
8. In the Lucas Lehmer Test for Mersenne Numbers p of the following form 2^p-1
   S₁ = 4, S_n = S_n-1²-2 (mod 2^p-1) for 2 ≤ n ≤ p-1.
   2^p-1 is being prime number if and only if S_p-1 = 0.
   
   In Prime95 for Windows and mprime for Unix and Linux, and then or software application uses with a some classified syntax is being on to
   S₂ = 4 and S_p = 0 rather than instead of S₁ = 4 and S_p-1 = 0. Why?
   Not why not classically enough of all for though of even for though!
   
   Besides S₁ = 4 in the Lucas Lehmer Test, what other values for S₁ in the Lucas Lehmer Test would be valid?
   Is there being a proper test, trial algorithm - a some run away formula to find them out? By the way, why does it work out any way? Not, on the other hand!
   S₁ = 4 or S₁ = 10 would be suitable or S₁ = 3 for p ≡ 0 or 1 (mod 3)? Not always!
   Not why not classically enough of all for though of even for though!
9. A some Mersenne prime number exponent p of the following form 2^p-1
   is said to be in equivalence class of (+) if and only if S_p-2 ≡ 2^(p+1)/2 (mod 2^p-1).
   and (-) if and only if S_p-2 ≡ (2^p-1)-2^(p+1)/2 (mod 2^p-1).
   What is being the significance of their own difference if any?
   Different starting point value of S₁ leads out to different (+) and (-) equivalence class distribution of the same and different elements. Right?
   Is their own probability or likelihood distribution of their own occurrence being sticking out just on to 50% - 50% ratio proportion - range limit bound - level scale rate?
10. Lucas Lehmer Test:
    S₁ = 4, S_n = S_n-1²-2 for n ≥ 2.
    In general, S_n = (2+√3)^2[sup]n-1[/sup]+(2-√3)^2[sup]n-1[/sup].
    Observe that S₄ ÷ (2 × 31) = 607, that is being a some Mersenne prime number exponent of the following a some form 2^p-1.
    Observe that S₆ ÷ (2 × 127) = 7897466719774591, a prime number so that, such that 2^{7897466719774591}-1 has been got with no small prime factors at all, not even (2^{7897466719774591}+1)/3!
    Could be that - 7897466719774591 a some Mersenne prime number exponent of the following a some form 2^p-1 at all?
    Only time can tell!
    And then or - is being that 7897466719774591 a some Wagstaff prime number exponent of the following a some form (2^p+1)/3 at all?
    Only time can tell!
    (2⁶⁰⁷+1)/3 is not being a some prime number at all!
    607 is not being a some Wagstaff prime number exponent of the following a some form (2^p+1)/3 at all!
11. Could we be able on to factor a Mersenne composite number - some prime number exponent from last iteration of Lucas Lehmer Test?
    If we know the period with which S_n repeats (mod q), where 1 < q < 2^p-1 is being a prime number and some proper divisor of 2^p-1, with different periods of modulo of other prime factors for 2^p-1 for S_n, then
    and if S_a ≡ S_b (mod q) is being smallest of repetitions, b-a is being the period with which S_n repeats (mod q), then
    we could be able on to extract a some factor of 2^p-1 by using GCD(S_b-S_a, q) - be being that way!
    Not a some useful technique although! If S₁ = (2+√3) is being an element of finite field GF(q²), then Lucas Lehmer test iterations would be like computing with S₂ = (2+√3)², S₃ = (2+√3)⁴, S₄ = (2+√3)⁸, etc. We cannot compute with S_2n from S_n for faster calculation of period, and factoring with 2^p-1 immediately. 2^p-1 is being prime number if and only if integer part of S_p-1 = (2+√3)^2[sup]p-2[/sup] ≡ 0 (mod 2^p-1).
    Does any one know of effective and then or efficient trial test algorithm or a some run away formula for computing the period with which Lucas Lehmer Test iteration repeats (mod q) where q is being a some prime number?
    

    p=2047
    f(a,b)=[a^2+3*b^2,2*a*b]
    g=Mod([2,1],p);h=Mod([7,4],p);i=1;while(g!=h,g=[2*g[1]+3*g[2],g[1]+2*g[2]];h=[2*h[1]+3*h[2],h[1]+2*h[2]];h=[2*h[1]+3*h[2],h[1]+2*h[2]];i++);print("Repeating period: "i)
    Of the following form - of the following form - indeed obviously rather than instead of - indeed obviously rather than instead of - also too - also too - also too - also too - of of just that is being - of of just that is being - and then or - with in only certainly - that which it that - very much all most - away out up - be being by using that way - away out up off down my own - of for from front frontier - with in with out with up with away with off with down with my with own - up on over in to - with with with with with with with with - in out up away off down my own - up on over in to - of of just that is being - of of just that is being - also too - also too - also too - also too - indeed obviously rather than instead of - indeed obviously rather than instead of - of the following form - of the following form!

    As the prime number q is being increasing, the period with which the Lucas Lehmer Test iteration repeats (mod q) also increases exponentially too, making it out that is being computationally out of reach for by using of computing power! Be being that way - for that which is being of exponential running time also too!
    Here is being for Tortoise Hare algorithm of PARI/GP language for programming of code!
    

    n=59
    g=Mod(4,2^n-1);h=Mod(14,2^n-1);i=1;while(g!=h,g=g^2-2;h=h^2-2;h=h^2-2;i++);print("Repeating period: "i);k=Mod(4,2^n-1);l=Mod(4,2^n-1);for(j=1,i,l=l^2-2);e=1;f=i+1;while(k!=l,a=k;k=k^2-2;b=l;l=l^2-2;e++;f++);print("a, b: "e", "f);print("S(a), S(b): "lift(a)", "lift(b));print("Factor: "gcd(lift(a)-lift(b),2^n-1))
    Right now that by right now that itself at this very moment variably - right now that by right now that itself at this very moment variably - right now that by right now that itself at this very moment variably - right now that by right now that itself at this very moment variably - right now that by right now that itself at this very moment variably - right now that by right now that itself at this very moment variably - right now that by right now that itself at this very moment variably - right now that by right now that itself at this very moment variably - given fixed - given fixed - given fixed - given fixed - given fixed - given fixed - given fixed - given fixed!

12. This is being some useless, not brilliant idea on to factoring a some Cunningham number of the following form (bⁿ-1)/(b-1) and then or (bⁿ+1)/(b+1):
    
    GCD(bⁿ-1, c²ⁿ-1) for a some integer c ≠ b.
    And then or GCD(bⁿ+1, c²ⁿ-1) for a some integer c ≠ b.
    
    GCD(2⁴⁰⁹⁶+1, 14⁴⁰⁹⁶+1).
    GCD(2⁵⁹-1, 659⁵⁹+1).
13. 1. What is the expected number of prime factors of a given N-digit number?
    2. What is the expected power of 2 for σ(x) where x is a N-digit number?
    3. What is the expected power of 2 for σ(x)-x where x is a N-digit number?
    4. What is the expected power of 2 for φ(x) where x is a N-digit number?
    5. What is the expected power of 2 for x-φ(x) where x is a N-digit number?
    6. What is the probability that a given N-digit number is prime?
    7. If x is a N-digit number, then what is the probability that (bⁿ-1)/(b-1) and then or (bⁿ+1)/(b+1) is prime?
14. (G. L. Honaker, Junior.) Observe that 61 divides with (67 × 71) + 1.
    61, 67, 71 are being consecutive prime numbers.
    Are there being an other 3 consecutive prime numbers so that, such that the same condition thing holds out?
    Could that be possible any way also too?
    By the way, why does it work out? Not, on the other hand! Not always also too!
15. Attaining out, Wieferich Prime Numbers: for just the values of 2^p-1-1 ≡ 0 (mod p²) at all also too.
    Obtaining out, Wilson Prime Numbers: for just the values of (p-1)!+1 ≡ 0 (mod p²) at all also too.
    Looking out, Wall Sun Sun Prime Numbers: for just the values of F_p-(p|5) ≡ 0 (mod p²) at all also too.
    Seeking out, have other people been got out considering out with in only certainly for just the values of that which 3^p-1-1 ≡ 0 (mod p²) at all also too?
    What are they being - could they be called as out - that it known as out? b^p-1-1 ≡ 0 (mod p²) for just the values of b ≥ 2 at all also too?

[Raman]

Of these stuff, different things belong to different parts of this mersenne forum, nevertheless for pondering thoughts right now, placing all of them over a some, one single part of this mersenne forum piece over here by - combined thread post - where by.
I think so.

1. Human beings first landed on surface of Moon on Sunday 20 July 1969. Can some one be able to predict when human beings will first land on surface of Mars? At least the year? My guess is that year being Sunday 16 September 2035. That will be one of the closest approaches of Mars to Earth during that cycle and taking time till equipments for such tasks and missions are being well developed. Earth takes 365.2422 days to revolve around the Sun and Mars takes 686.9 days to revolve around the Sun. At opposition of Mars from Earth, after how many days will they meet again back right now? On the other hand, Venus is being too hot for human beings up to land on its own surface. Thick atmosphere of CO₂. 93 times atmospheric pressure up on surface of Earth. Strong run away greenhouse effect - over there by - warming up its own surface out away - where by.
   Spacecrafts took about 3 days to travel from surface of Earth to surface of Moon. Will it take 8 months for spacecraft to travel from surface of Earth to surface of Mars? I had heard that astronauts are being practising under Mars like atmospheric conditions in the island of Hawaii before departure. Do you think that it will ever become a reality for spacecraft to travel at speeds approaching the speed of light? Will it ever become a reality to travel between two different antipodal points on surface of Earth in matter of even minutes or even seconds? What prediction do you think so of? Why?
2. Do you know that zh should not be used in Tamiɻ language for ழ sound? According to the International Phonetic Alphabet, the proper symbol for that pronunciation letter is being ɻ ligature. Zh sound clearly stands out for the russian Ж letter always, but that ɻ sound is not same as that ʒ sound. ɻ symbol is being very much appropriate in my opinion, close to the letter of l, a hook symbol denotes away, indicates away with a slight modification to that sound pronunciation.
3. In phonetics, is it being possible that a certain person with a some, problem in the throat, larynx, voice box while speaking is being able to pronounce with a some sounds, where as, but not (dumb enough for) other sounds? Why? Similar to that of colour blindness. Blue yellow colour blindness is being very much rarer than red green colour blindness. Why?
4. 1. Is Puerto Rico going to become United States of America's 51^st state?
   2. Name of Nubia for South Sudan would be fancier! Democratic Republic of Congo for Zaire and Serbia, Montenegro and Kosovo for Yugoslavia why would the names get changed in to lesser fancier names? Why is Antarctica not being shown off in maps some times?
   3. In Chennai, after fly overs at Vadapaɻani Metro Railway Station, fly overs at Porur Junction, Vandalur Zoo, Kelambakkam junction, Velachery Vijaya Nagar Bus Stand, Tidel Park, SRP Tools junction and Jayanthi Theatre Junction would be very much useful enough up to the people. These areas are being frequently often stranded by using traffic jams. And then be being that are projects under construction are already being started away or began out up? Why not?
   4. Amaravathi is being the new proposed capital of Andhra Pradesh state of India after it was being split up from Telangana State up on Monday 2 June 2014. Could large cities like Visakhapatnam serve as the capital of this state very much? Why not? Or large towns like Guntur or Vijayawada? Amaravathi is being already built up over the outskirts of just these places!
   5. I would like to suggest up a train service operating over Old Mahabhalipuram Road. Or operating over East Coast Road would be very much better off? Why not?
   6. I would like to also suggest up implementing ₹ 1000 currency notes money symbol too. This was already being implemented a some time long ago! Why not? This ₹ sign was being officially introduced up on Thursday 15 July 2010.
   7. Suggest Jasper named hostel at National Institute of Technology, Tiruchirappalli also too.
   8. Why signals show with up arrow for going straight? Also does it mean to drive up wards in to the skies and is it being possible any way too?
   9. Two other places / people with in same names should be discouraged only certainly, as far as possible.
      Right now that - immediately - as soon as possible - by right now that itself - at this very moment variably - up on over in to - given fixed - intermediate - in between.
   10. Right now that Sun passes every day at over head point at zenith point with in the continent of Africa only certainly - not Europe and then or Antarctica at all!
   11. In Tamil Nadu, vehicle number plates letters do not use letters of I and O. G is being reserved for government vehicles and N is being reserved for buses - public transport and then or other government people transportation vehicles.
   12. In serial numbers with letters, other people seldom use letters of I and O.
   13. Letters of a some H, N, R, T, Z are being under a some problem for me.
   14. Working out just at a some ending times do not yield a some problem for me - at all - also too - of for of for!
   15. Many aeroplanes and movie film theatre shows and other services skip with the number 13.
   16. Friday the 13^th is being considered unlucky.
   17. In Mersenne Forum, list tag supports bullets, alphabets, numbers, roman numerals, both of upper case list tag and of lower case list tag and both!
   18. This mersenne forum must should consider continuing on to list serial numbers in to next subsequent thread post numbers.
   19. Is our own surface of Earth being losing water to space due to run away planet evaporation?
   20. What are being volatile chemical elements?
   21. Why are those areas classified being near by on to equator humid enough, near by on to tropics of cancer and capricorn arid enough, near by on to temperate middle latitudes wet enough, near by on to poles dry enough?
   22. Why are very much being all most all subduction zones located near by on to coast lines of continents and islands and peninsulas?
   23. My post counts have been suddenly increased by over a 200 during the 2½ year period that I had been inactive. Why? Due to the opening of chess subforums, they have now been included into post counts as like that which they were not before?
       Mersenne forum post counts right now shows out as prime factors or like some arbitrary base representation! What ever else new representation is being possible that ever which ever has been available a way a way? Sake purpose process case. Random = custom - beginning out?
       Very much tempting enough to starting out for the running up executing out trying attempting admitting attending with in computation of an Aliquot Sequences starting out, beginning out from these numbers very much, right now!
   24. Aliquot Sequence 1578 Iteration Number 7564. 2¹⁰⁷-1 = 162259276829213363391578010288127 substring 1578.
   25. 18 years old - age of majority - 6574 days old - 4 leap years included up on to getting out old involved.
       6575 days old - 5 leap years included up on to getting out old involved.
       And then or - 6573 days old - century years sandwich oldness - 4 leap years included up on to getting out old involved.
       6574 days old - century years sandwich oldness - 5 leap years included up on to getting out old involved.
       3⁸ days old = 6561 days old = 9⁴ days old = 81² days old = 6561¹ days old = 10000₉ days old = 100000000₃ days old = 100₈₁ days old = 10₆₅₆₁ days old ≃ 17.96 ~ 17.97 years old.
       10000 days old ≃ 27.37 years old.
       Saros sequence - series cycle ≈ 18 years 10 / 11 days old.
       Metonic cycle - sequence series ≈ 19 years 0 days old.
       Had been got out - seeking out - looking out - obtaining out - attaining out!
   26. Mornings are being cooler, evenings are being warmer.
   27. Once written out - one single time - time period frame duration - time times know known
   28. Solving out NP-complete complexity class problems that which essentially means out that it captures out essence of solving out all problems up on to with in NP complexity class only certainly - of for of for!
   29. Just an other point always - called as out - known as out - a some - very much all most all
       Accelerating of education system could be encouraged out up on to - and then or considered out - a some - of for of for - counted out with in only certainly - very much all most all!
       Those people guys who ever will be finishing off with in their own education system up on to very much all most all at birdie and then or earlier times will be a some benefitting maximum gain - only certainly - of for of for!
       Those people guys who ever will be finishing off with in their own education system up on to very much all most all at par and then or usual times will be earning a some maximum neither profit nor loss - only certainly - of for of for!
       Those people guys who ever will be finishing off with in their own education system up on to very much all most all at bogey and then or later times will be suffering a some maximum loss - only certainly - of for of for!
       A some - limit range bound level - rate ratio scale proportion - exists out - very much all most all!
   30. 29 - 2 not joining all together! 29 - February not merging all together!
   31. Incoherent.
       Reference - Glossary of Data - Data Under Preparation - Less Important Data - Off Topic Data - Archive of Data.
5. Will robots eventually become a reality to automate work outs of human beings? When? Will singularity occur? Other people predict that in year of 2045, robots will dominate human beings work outs, artificial intelligence will far away surpass that of human beings! Robots will compete with human beings starting from walking out up on roads, beginning with by using a some techniques - be being that way!
   In general, automating computer software - hardware programming code - is being RE-complete - and then or AI-complete secret password? Just similarly on to Hilbert's 10^th problem style out!
6. Have other people been already considering about rain water harvesting for drinking water? Here is a some method for getting pure and clean drinking water! By condensing humidity in the air with a some cold water bottle or ice cubes or cooling agents or some thing like that, we would be able to get with lot of pure and clean drinking water for free! But that the amount of water vapour in to the air might be lesser, lower at places closer in to the poles, away from the equator, where by it would be very much higher over here by very much!
7. One way of preventing thieves and robbers or bandits and dacoits in to house is by using spikes up on the entrance and spikes will be gone back into the down holes only if a some correct password or a some secret code is being entered up away out!
8. Once a star dies, it will eject material in to forming supernova explosions that will give rise to birth of newer lesser massive stars, just like women giving rise to birth of children, children later growing up from then on wards. Will lesser massive stars grow out away? Right now, this chain reaction will continue for 10¹⁴ of years till or until stars are not being massive enough to fuse hydrogen in to helium - the temperature that is being needed away for this reaction out up will not arise! Or could universe crunching and rebounding be a cyclic process or phenomenon? What is being the shape of our own universe? Could Big Bang process or phenomenon be a some star death and supernova explosion of a some very much massive enough star?
   Only 10% of previous star material certainly survives in to next generation but that rest are being emitted away in to nebula out up. All though, of even though super nova explosions last only for certainly with in smaller periods of time when compared to entire life span of the star. When do you predict that Betelguese star will become a supernova. At least what year range limit a some level bound? We are being made up on of star dust material of dead stars long ago. Our own surface of Earth is being made up on of star dust, with atmospheric layers of nitrogen, oxygen and traces of argon and carbon di oxide, with crust layers of aluminium and silicon, with mantle layers of magnesium and silicon, with core layers of iron and nickel, with layers right now.
9. Human beings live up to the age of 82 years up on average scale a some rate ratio proportion. Their own maximum productivity will be, is being at the age of 26 years, right now. Right now, compare this with against chemical elements! All chemical elements with atomic number > 82 (lead) are all being radioactive and unstable away out up! The maximum binding energy of chemical elements occurs with for atomic number of 26 (iron) with 56 total number of protons and neutrons, originally produced in the star with an atomic number of 28 (nickel). Formations of chemical elements heavier than iron in very much massive enough stars only possible certainly requires absorption of energy from heat energy (endothermic reaction), but while where as those lighter than iron requires release of energy as heat energy (exothermic reaction) right now - right now!
   In our own surface of Earth's atmosphere, argon-40 was all mostly being formed away out up from the β+ decay of potassium-40. Argon-36 isotope is being the very much all most preferred isotope of argon that is being formed up with in all the stars out away - very much all most. Chemical elements of with a some odd number as atomic number are being rarer than chemical elements of with a some even number as atomic number! Lesser favourable against with formation for in stars right now - right now! Exceptions include atomic number of 1 (hydrogen) that is being the very much all most abundant chemical element with in the universe - formed in abundance in Big Bang along with atomic number of 2 (helium) chemical element and atomic number of 4 (beryllium) chemical element that is being very much all most rarely formed with in our own surface of Earth produced only certainly by using cosmic ray spallation, rather than instead of of just that is being destroyed with in star formation process or phenomenon during stellar nucleosynthesis only certainly, along with atomic number of 3 (lithium) chemical element and atomic number of 5 (boron) chemical element - be being that way (Oddo - Harkins rule of of just that is being)!
   On the other hand, very much all most all terrestrial origin helium consists of helium-4 isotope, so that, such that, a some, any sample of of just that is being rich with in helium-3 isotope is being likely to be of extra terrestrial origin very much only certainly - according on to all most all probability and then or likelihood rate scale ratio proportion - level limit range bound - up on over in to.
   Independent of production with in mechanism of star formation and then or stellar nucleosynthesis only certainly - of the following form - given fixed - way method process procedure and then or phenomenon methodology mechanism - right now that by right now that itself at this very moment variably!
   What causes us in to growing?
10. Greek script alphabet has been got with in 24 letters only certainly.
    Latin script alphabet has been got with in 26 letters only certainly.
    Cyrillic script alphabet has been got with in 33 letters only certainly.
    Exactly 24 / 26 / 33 fixed points of of just that is being given away out up.
    Keeping on to adding with in a some points till until then only certainly - range limit bound level - ratio rate scale proportion.
    3 × 3 magic square good on to using with in numbers from 1 up to 9 digits only certainly.
    5 × 5 jigsaw sliding puzzle good on to using with in 24 Greek script characters of of just that is being given fixed mapped away out up only certainly.

[Raman]

I had to add up on with in a some of these prime factors for in to factordb.com web site page - for only of certainly ≤ 10000000 digits - limit range bound level - rate ratio scale proportion - exists out!
Factordb.com web site page does not accept with in a some of larger factors for ≥ 10000000 digits - exists out - for only of certainly!
Factordb.com web site page 10000000 digit limit exists out - parallel execution queries limit exists out - sequential input queries limit exists out!
Factordb.com web site page does not mark out larger known found out prime numbers and then or probable prime numbers called as out - known as out!
Been got out - seeking out - looking out - obtaining out - attaining out!

With in this following PARI/GP command only certainly and then or script
you could be able on to getting away with the very much all most all prime factors of 2^p-1 and then or (2^p+1)/3 both simultaneously - a some - of just the following given fixed form!
And then or - called as out - known as out - namely - that which it that!
p = (2⁷⁹+1)/3

p=(2^79+1)/3
forstep(q=2*p+1,10^50,2*p,if(Mod(4,q)^p==1,print(q)))
Of for of for - with in only certainly - away up out down off my own - that ever which ever a way a way ever - by using be being that way - that which it that. And then or - and then or - of for from front frontier - right now that by right now that itself at this very moment variably. Very much all most all - away up out down off my own - that ever which ever a way a way ever - by using be being that way - that which it that. Once written out - one single time - time period frame duration - time times know known. A some - a some - a some - a some - a some - a some - a some - very much all most all. Away up out down off my own - that ever which ever a way a way ever - by using be being that way - that which it that. Once written out - one single time - time period frame duration - time times know known. Called as out - known as out! Once written out - one single time - time period frame duration - time times know known. And then or. Namely - that which it that. A some - a some - a some - a some - a some - a some - a some - very much all most all. Just an other point always - called as out - known as out - a some - very much all most all. A some - limit range bound level - rate ratio scale proportion - exists out - very much all most all!

Right now that I am trying it out ECM curves up on over at following four numbers (2⁸¹⁹¹+1)/3, (2¹⁹⁹³⁷+1)/3, (2¹¹⁰⁵⁰³+1)/3, (2⁵²⁴²⁸⁷+1)/3 with in higher bounds on a four core micro processor.

Double Mersenne

23-1 is prime.
27-1 is prime.
231-1 is prime.
2127-1 is prime.
28191-1 has the factors: 338193759479, 210206826754181103207028761697008013415622289.
2131071-1 has the factors: 231733529, 64296354767.
2524287-1 has the factors: 62914441, 5746991873407, 2106734551102073202633922471, 824271579602877114508714150039, 65997004087015989956123720407169.
22147483647-1 has the factors: 295257526626031, 87054709261955177, 242557615644693265201, 178021379228511215367151.

Wagstaff Mersenne

(23+1)/3 is prime.
(27+1)/3 is prime.
(231+1)/3 is prime.
(2127+1)/3 is prime.
(28191+1)/3 is composite.
(2131071+1)/3 has a factor: 2883563.
(2524287+1)/3 is composite.
(22305843009213693951+1)/3 has a factor: 1328165573307087715777.

If the new Mersenne conjecture of of just that is being true, and then or 2^{2305843009213693951}-1 - is being that - should be that - a some composite number - indeed obviously rather than instead of!
What is being the probability and then or likelihood that at least one of the four numbers, namely - that which it that - 2^{2305843009213693951}-1, then, 2^{618970019642690137449562111}-1, or, 2^{162259276829213363391578010288127}-1, and, 2^{170141183460469231731687303715884105727}-1, - could be that - must be that - ought on to be that - a some prime number?
Once written out - one single time - time period frame duration - time times know known
Namely - that which it that

If the new Mersenne conjecture of of just that is being true, and then or (2^2147483647+1)/3 - is being that - should be that - a some composite number - indeed obviously rather than instead of!
If the new Mersenne conjecture of of just that is being true, and then or 2^{768614336404564651}-1 - is being that - should be that - a some composite number - indeed obviously rather than instead of!

What is being the probability and then or likelihood that at least one of the four numbers, namely - that which it that - (2^2147483647+1)/3, then, (2^{618970019642690137449562111}+1)/3, or, (2^{162259276829213363391578010288127}+1)/3, and, (2^{170141183460469231731687303715884105727}+1)/3, - could be that - must be that - ought on to be that - a some prime number?
What is being the probability and then or likelihood that at least one of the four numbers, namely - that which it that - 2^{56713727820156410577229101238628035243}-1, then, (2^{56713727820156410577229101238628035243}+1)/3, or, 2^{170141183460469231731687303715884105727}-1, and, (2^{170141183460469231731687303715884105727}+1)/3, - could be that - must be that - ought on to be that - a some prime number?
What is being the probability and then or likelihood that at least one of the four numbers, namely - that which it that - 2^715827883-1, then, 2^{2932031007403}-1, or, 2^{768614336404564651}-1, and, 2^{845100400152152934331135470251}-1, - could be that - must be that - ought on to be that - a some prime number?
What is being the probability and then or likelihood that at least one of the four numbers, namely - that which it that - (2^715827883+1)/3, then, (2^{2932031007403}+1)/3, or, (2^{845100400152152934331135470251}+1)/3, and, (2^{56713727820156410577229101238628035243}+1)/3, - could be that - must be that - ought on to be that - a some prime number?

Mersenne Wagstaff

23-1 is prime.
211-1 = 23 × 89.
243-1 = 431 × 9719 × 2099863.
2683-1 = 1367 × 434836499112609694795723958417048861299768144283442662402095922180462812746769 × 67513796971703570854592232797421324116119881147340327278928245456644619398078155616494185719845536064262986241999463764460809.
22731-1 has the factors: 93968249, 5235895818143, 697275709026751, 563358792984278565516774152727223543227673.
243691-1 has the factors: 87383, 1398113, 4690767254460090160943, 1787363373488812416764791.
2174763-1 is composite.
22796203-1 has the factors: 5592407, 17017419583182311, 23349981773942355169801.
2201487636602438195784363-1 has a factor: 14549422239062062117588852231.

Double Wagstaff

(23+1)/3 is prime.
(211+1)/3 is prime.
(243+1)/3 is prime.
(2683+1)/3 = 1676083 × 26955961001 × 296084343545863760516699753733387652635366098889116410731661924253563729059085336779932810899819313612925255002666691226800507277398580985624625950496168983999760414855301693388419156899841.
(22731+1)/3 has the factors: 67399191280564009798331, 2252735939855296339250682011.
(243691+1)/3 has a factor: 349529.
(2174763+1)/3 has a factor: 173085275201.
(22796203+1)/3 has a factor: 129469791307.
(2768614336404564651+1)/3 has a factor: 3290547117383710719111443.
(2201487636602438195784363+1)/3 has the factors: 183756724581423634555339057, 101874969893105185923314913883.

Mersenne Fermat

23-1 is prime.
25-1 is prime.
217-1 is prime.
2257-1 = 535006138814359 × 1155685395246619182673033 × 374550598501810936581776630096313181393.
265537-1 has the factors: 513668017883326358119, 883852340565787164089923172087.

Wagstaff Fermat

(23+1)/3 is prime.
(25+1)/3 is prime.
(217+1)/3 is prime.
(2257+1)/3 = 37239639534523 × 518144156602508243009 × 4000659204579114753312310878847043394855313.
(265537+1)/3 has a factor: 13091975735977.

Larger candidate numbers - fewer prime factors - pumped out - not - not - smaller - not - not - several - very much all most all - a some - and prime then composite or unique! Shorter candidate numbers - many prime factors - pumped out - not - not - bigger - not - not - lesser - very much all most all - a some - and prime then composite or unique! Cunningham Tables numbers candidates! Fibonacci numbers, Lucas Numbers, Homogeneous Cunningham Numbers and other twisted additive or multiplicative groups like these things. As ≠ like last final ultimate next previous letter character alphabet digit number numeral cardinal ordinal stuff.
As ≠ like last final ultimate next previous letter character alphabet digit number numeral cardinal ordinal stuff. Fibonacci numbers, Lucas Numbers, Homogeneous Cunningham Numbers and other twisted additive or multiplicative groups like these things. Cunningham Tables numbers candidates! Lower candidate numbers - a lot of prime factors - pumped out - not - not - greater - not - not - sparser - very much all most all - a some - and prime then composite or unique! Huger candidate numbers - rarer prime factors - pumped out - not - not - tinier - not - not - a plenty of - very much all most all - a some - and prime then composite or unique!

[xilman]

Originally Posted by Raman

I would like to give a few of my suggestions to make GIMPS Project better.[LIST=1][*]At the first place, I would suggest that Prime95 would use a B2 bound of GMP-ECM default and not 100 × B1.
Though it is not of much of difference for smaller factors, but as the factors get bigger, this would be a serious bottleneck.
For small Fermat numbers and Mersenne numbers without any known factors or are incompletely factored, see the difference between Prime95 and GMP-ECM.
For example, consider at the 65 digit level, with B1 = 850000000, GMP-ECM requires only 69471 ECM curves while Prime95 requires over 360000 ECM curves.
What a waste of computing power!

[deleted material]

Please fix this serious issue immediately, as soon as possible!

GMP-ECM uses a very large amount of memory in its second stage to allow it to use much bigger B2 bounds for relatively little computation. It can get away with it because GMP-ECM is typically used for small numbers.

If Prime95 used higher B2 bounds it would have to use either prohibitively large amounts of memory or much more computation in the second stage than it now does. In the latter case, relatively straightforward analysis shows that the computational costs of the increaded B2 bound greatly outweighs those incurred in running more curves at a lower B2 bound.

Summary: it is not a serious issue which needs fixing immediately. It is something which has already been given careful thought.

[Raman]

One hour after wards posting, post edit timer expired, 76^th result.
76. 2^1400251-1 has a factor: 2190658775806151479217.
Entered all computational results, prime factors up in to with in factordb.com web site page only certainly.

Some Prime Net assignments carry up on to 1 ECM curve with in running away only certainly - when assigned 3 ECM curves executing away.
Personal computer raw draft preparation, mersenne forum appeared thread posts up on to keep with in synchronization only certainly!

Originally Posted by Raman

Reason For Editing: None.
Reason For Editing: No reason was specified.
Reason For Editing: (Un)title(d).
Reason For Editing: Go Advanced.

Up on to with in only certainly!

Wait waitist out before ≠ after up posting thread ≠ post.

[GP2]

Originally Posted by Raman

I would like to give a few of my suggestions to make GIMPS Project better.[LIST=1][*]In Prime95 p-1 and ECM menu, why does the exponent still remain as 1061? 2¹⁰⁶¹-1 has already been completely factored over 4 years ago by now, and it could be changed to some thing like 1277. How ever I believe that factoring 2¹²⁷⁷-1 is unlikely to produce any productive results, even when B2 has been set to GMP-ECM default range level.

It's true that if you select Advanced → ECM... or Advanced → P−1... from the Prime95 menu, the dialog box suggests using 1061, an exponent which was fully factored in August 2012. Maybe in the next version it could be changed to 1277, as you suggest. To be useful, the B1 bound would have to be greatly increased to 800000000 for ECM, and maybe somewhat lower for P−1.

However the full ECM history for M1061 indicates that no ECM tests have been reported for this exponent after August 2012 (i.e., nothing beyond whatever stragglers were already in progress at the time the factor was found). So it probably wouldn't make much practical difference.

Originally Posted by Raman

Why does ECM Progress web site page still contain exponents like 1409 which are being completely factored away right now?

That's a good question. A quick check shows that the following proven-fully-factored exponents are present in the ECM Progress report: 1409 2087 2243 2381 10169 57131 58199 63703, and in addition it seems that nearly all the probably-fully-factored exponents are also present (i.e., 82939 86137 86371 106391 130439 136883 157457...), although 87691 is omitted.

However, M1409 was only fully factored in February of this year, and the ECM history shows no further ECM tests have been done since then. Still, for the sake of consistency it should omit all the fully-factored exponents and not just some or most of the small ones.

The problem is that only mersenne.ca records information about fully-factored and probably-fully-factored exponents (there are 304 of them known so far) while Primenet (mersenne.org) doesn't store that information at all.

Originally Posted by Raman

Why does some stupid guys run ECM curves, p-1 and p+1 on prime number candidates like 2¹¹²¹³-1, 2¹⁹⁹³⁷-1, 2²¹⁷⁰¹-1, 2²³²⁰⁹-1, 2⁴⁴⁴⁹⁷-1, 2⁸⁶²⁴³-1, etc.?

The server doesn't hand out those exponents. Probably those guys manually set up an entire range of exponents and forgot to filter out the Mersenne primes, and the fully-factored and probably-fully-factored exponents.

Originally Posted by Raman

Could mersenne.org web site page in the future times record with in the σ value for every ECM curve ran away or executed away whose successful factor is being found out some times by using the other people only certainly up right now?

Why would storing the σ value of a successful curve be useful? All we care about is the factor itself.

The σ value does get reported in the results line that is sent to Primenet, so if absolutely necessary Madpoo could probably dig it out of some log file.

Originally Posted by Raman

Once I had observed that in the Factors Found page, TJAOI was able to find out a continuous sequence of factors in strictly increasing order that divide some Mersenne composite number 2^p-1 for some prime number p ≤ 10⁹.
Does any one over here by has got an idea of what algorithm or script is he being running away or executing away?

There is an entire (long) thread about user TJAOI, speculating about methods and goals. As far as I know, no one from this forum has communicated directly with this user.

Originally Posted by Raman

It would be good if a some of the larger Mersenne composite number candidates get some p+1 and ECM curves ran away / executed away after wards of trial factoring and then p-1, or before sending them in to a first time Lucas Lehmer trial test run away / executed away.

I believe ECM is much too slow to be useful when exponents are of that size. In practice, we only use it to find factors of smaller exponents, and all of those have already been factored or LL tested.

P−1 testing is particularly efficient for testing Mersenne exponents because in effect the "minus 1" is cancelled out by the "plus one" of 2kp + 1. I don't think P+1 has any particular advantage for Mersenne exponents and in any case mprime doesn't implement it. Although GMP-ECM implements P+1 testing, it is not specifically tuned for testing Mersenne numbers quickly. I wonder if anyone has ever used P+1 testing successfully to find a factor of a Mersenne number (apart from maybe very small exponents).

[GP2]

Originally Posted by Raman

Secondly, I would suggest that PrimeNet server assigns tasks with ECM on small Mersenne numbers with not only on Mersenne composite numbers with no known factors but also which are partially factored.

Anyone can do this manually on their own initiative, looking for additional factors of small exponents which already have known factors. I myself started doing this recently. The difficulty is only in creating the appropriate "known factors" string at the end of each line, but I wrote a Python script to automate this.

However, this is just for fun. It doesn't really align with the goals of the project.

For exponents that have been LL tested composite but have no known factors, it is still useful for the project's goals to look for a (first) factor, for two reasons. First of all, there are a bunch of older machines which lack the horsepower to do anything else, but they can still run a few ECM curves. This keeps users involved who might otherwise be unable to participate, and who knows, someday they might get a new fast computer.

Second, it is qualitatively much better to possess a factor than just an LL test. If someone supplies you with a factor, it is trivial to verify it for yourself. It takes only a few microseconds of computing time, in fact you can verify the entire database of tens of millions of factors in one shot, in a minute or two. On the other hand, if someone only supplies you with an LL residue, then it is much more laborious to verify it for yourself. The time to run an LL test on M_p is O(p² log p). So in the meantime you have to worry whether the user is reliable (was the test faked?), whether the software is reliable (does it have rare bugs?), and whether the hardware is reliable (are there memory errors?).

So long as there are Mersenne exponents with no known factors, those will always be higher priority for the project's goals than Mersenne exponents that already have known factors.

[GP2]

Originally Posted by xilman

If Prime95 used higher B2 bounds it would have to use either prohibitively large amounts of memory or much more computation in the second stage than it now does.

Isn't it both?

[xilman]

Originally Posted by GP2

Isn't it both?

Not obviously so. Not obvious to me, anyway, but I may be quite wrong.