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 2016-09-26, 10:00 #1 Raman Noodles     "Mr. Tuch" Dec 2007 Chennai, India 3·419 Posts Prime95 - suggest using B2 bound = GMP-ECM default and other questions 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. 2158209-1 has a factor: 26496805856040782582748460014520511 21413017-1 has a factor: 2226694532490185824727 21320091-1 has a factor: 231827487452450337577 21085809-1 has a factor: 8381291184102382541497 21420109-1 has a factor: 85148369868485192804575096841 21306477-1 has a factor: 2886593156080095874622681 21264387-1 has a factor: 111421197216700721651761 21073647-1 has a factor: 45496627768358111930287 21297129-1 has a factor: 21635389012955382990854233 21249741-1 has a factor: 168691067410297172399521 21415957-1 has a factor: 49084592298749387589959 21205173-1 has a factor: 269867708244063132649 21335563-1 has a factor: 1059938762839012963385177 21110167-1 has a factor: 14693675794413895601977 21185697-1 has a factor: 103519616241334574089 21185319-1 has a factor: 187769929174579865730719 2680077-1 has a factor: 359004574534541931650318449 (σ = 1545834309296023) 21406857-1 has a factor: 76972723442421389411833 21103281-1 has a factor: 461658571972334163101951 21082533-1 has a factor: 1623069162917486699809 21001629-1 has a factor: 25980809010119884941817 2679669-1 has a factor: 63260117764832948321737 (σ = 5074446274686185) 21409633-1 has a factor: 4562270691505620280350953 21314161-1 has a factor: 200209995939544733759 21401767-1 has a factor: 208164200904253362898039 21309811-1 has a factor: 6452094516571781903870908529 21212769-1 has a factor: 22686603584990095872353327 21155919-1 has a factor: 2446997794177577666902847 2674483-1 has a factor: 1185762874913395945847 (σ = 8472293866228076) 21151041-1 has a factor: 98338208354439540871 21389149-1 has a factor: 1968838610224783811144921 21362349-1 has a factor: 4052494766327711626877807 2640483-1 has a factor: 35948758457297251395594732823879 (σ = 7385969443002972) 21238119-1 has a factor: 19398358097550480988759 21361609-1 has a factor: 8394136703834908414711 21381381-1 has a factor: 486078017602417074080489 2130547-1 has a factor: 740710078242573288550675295986147001 21352543-1 has a factor: 54914730373539406420367 21048361-1 has a factor: 317587687414956893560559 21062367-1 has a factor: 5715626615941288867913951 21209781-1 has a factor: 61574197954768721757880409 2614909-1 has a factor: 2157432123233208899520343 (σ = 5192865221372438) 21017043-1 has a factor: 2714087245272479000359 21403807-1 has a factor: 910040398878981948860243369 21151327-1 has a factor: 374909084903502844569223 21394089-1 has a factor: 12522193154648380174007 21049891-1 has a factor: 14792762552601957907766311 2619979-1 has a factor: 39952445419572244259873 (σ = 8735607385635865) 21244249-1 has a factor: 440414964773453095807 21176221-1 has a factor: 461610300972747494021593 21030219-1 has a factor: 3772589184204120078374871601 21226387-1 has a factor: 1074355317155260325278943 21062793-1 has a factor: 12513099136956335921312007791 21023769-1 has a factor: 121894953460427063046553 21196059-1 has a factor: 17663439686039008542085063 (σ = 1539397084932662) 21111949-1 has a factor: 131359229602139901573898435183 21328891-1 has a factor: 190486443247254756345034440671 21198261-1 has a factor: 186410816493618076522807 (σ = 933907529783551) 21058077-1 has a factor: 1155399346576072505554663 21295611-1 has a factor: 1538613486295112984311 21387681-1 has a factor: 7635519794781065554162223 (σ = 7915873887005268) 21293421-1 has a factor: 46804080444409785497993 21094623-1 has a factor: 7153958318333686172617 21409549-1 has a factor: 6884935783696551152303 21045487-1 has a factor: 12870043214611775340199 21196123-1 has a factor: 25412328569397020505047 21104017-1 has a factor: 10390283462108941247777 21396529-1 has a factor: 3430327700879658519722567 21269167-1 has a factor: 69389537933503942909169 21357901-1 has a factor: 4006306700470164867761 21177741-1 has a factor: 1393309518989601569849 21172207-1 has a factor: 3612667936408840580839 2683831-1 has a factor: 817118841184883531740073 (σ = 1237148984876076) 21342907-1 has a factor: 3583767036379540468879 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?
 2016-09-26, 10:02 #3 Raman Noodles     "Mr. Tuch" Dec 2007 Chennai, India 3·419 Posts 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. Can I have a Mersenne Forum badge for latest prime number found out 274207281-1? 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 (10p+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! 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? 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. 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? 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? Consider with Mersenne Numbers of the following form 2p-1, there are being plenty of prime numbers of the following form 2kp+1 but only a few of them would divide 2p-1 such that their own product ≤ 2p-1. Due to a theorem of Number Theory, a prime number of the following form 2kp+1 should indeed obviously divide with 2kp-1. So that, 2kp+1 should divide with 2p-enπi/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 2p-1 (when 2 is being quadratic residue (mod 2p+1), 2p+1 ≡ 1 or 7 (mod 8)), or divide with 2p+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!) In the Lucas Lehmer Test for Mersenne Numbers p of the following form 2p-1 S1 = 4, Sn = Sn-12-2 (mod 2p-1) for 2 ≤ n ≤ p-1. 2p-1 is being prime number if and only if Sp-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 Sp = 0 rather than instead of S1 = 4 and Sp-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! A some Mersenne prime number exponent p of the following form 2p-1 is said to be in equivalence class of (+) if and only if Sp-2 ≡ 2(p+1)/2 (mod 2p-1). and (-) if and only if Sp-2 ≡ (2p-1)-2(p+1)/2 (mod 2p-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? Lucas Lehmer Test: S1 = 4, Sn = Sn-12-2 for n ≥ 2. In general, Sn = (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 2p-1. Observe that S6 ÷ (2 × 127) = 7897466719774591, a prime number so that, such that 27897466719774591-1 has been got with no small prime factors at all, not even (27897466719774591+1)/3! Could be that - 7897466719774591 a some Mersenne prime number exponent of the following a some form 2p-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 (2p+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 (2p+1)/3 at all! 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 Sn repeats (mod q), where 1 < q < 2p-1 is being a prime number and some proper divisor of 2p-1, with different periods of modulo of other prime factors for 2p-1 for Sn, then and if Sa ≡ Sb (mod q) is being smallest of repetitions, b-a is being the period with which Sn repeats (mod q), then we could be able on to extract a some factor of 2p-1 by using GCD(Sb-Sa, 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 S2n from Sn for faster calculation of period, and factoring with 2p-1 immediately. 2p-1 is being prime number if and only if integer part of Sp-1 = (2+√3)2[sup]p-2[/sup] ≡ 0 (mod 2p-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? Code: 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! 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! 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). What is the expected number of prime factors of a given N-digit number? What is the expected power of 2 for σ(x) where x is a N-digit number? What is the expected power of 2 for σ(x)-x where x is a N-digit number? What is the expected power of 2 for φ(x) where x is a N-digit number? What is the expected power of 2 for x-φ(x) where x is a N-digit number? What is the probability that a given N-digit number is prime? 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? (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! Attaining out, Wieferich Prime Numbers: for just the values of 2p-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 Fp-(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 3p-1-1 ≡ 0 (mod p2) at all also too? What are they being - could they be called as out - that it known as out? bp-1-1 ≡ 0 (mod p2) for just the values of b ≥ 2 at all also too? Last fiddled with by Raman on 2016-09-26 at 11:01 Reason: Reason For Editing.
 2016-09-26, 10:04 #5 Raman Noodles     "Mr. Tuch" Dec 2007 Chennai, India 100111010012 Posts 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 2p-1 and then or (2p+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 Code: 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 Code: 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 Code: (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 22305843009213693951-1 - 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 - 22305843009213693951-1, then, 2618970019642690137449562111-1, or, 2162259276829213363391578010288127-1, and, 2170141183460469231731687303715884105727-1, - 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 (22147483647+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 2768614336404564651-1 - 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 - (22147483647+1)/3, then, (2618970019642690137449562111+1)/3, or, (2162259276829213363391578010288127+1)/3, and, (2170141183460469231731687303715884105727+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 - 256713727820156410577229101238628035243-1, then, (256713727820156410577229101238628035243+1)/3, or, 2170141183460469231731687303715884105727-1, and, (2170141183460469231731687303715884105727+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 - 2715827883-1, then, 22932031007403-1, or, 2768614336404564651-1, and, 2845100400152152934331135470251-1, - 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 - (2715827883+1)/3, then, (22932031007403+1)/3, or, (2845100400152152934331135470251+1)/3, and, (256713727820156410577229101238628035243+1)/3, - could be that - must be that - ought on to be that - a some prime number? Mersenne Wagstaff Code: 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 Code: (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 Code: 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 Code: (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. Code: 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! Last fiddled with by Raman on 2016-09-26 at 10:35 Reason: Wrapped code tags to keep width of window in check.
2016-09-26, 10:16   #6
xilman
Bamboozled!

"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

132×67 Posts

Quote:
 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.

2016-09-26, 11:29   #7
Raman
Noodles

"Mr. Tuch"
Dec 2007
Chennai, India

3×419 Posts

One hour after wards posting, post edit timer expired, 76th result.
76. 21400251-1 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!

Quote:
 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.

Last fiddled with by Raman on 2016-09-26 at 11:54 Reason: Reason For Editing: None, No reason was specified, (Un)title(d), Go Advanced.

2016-09-26, 12:09   #8
GP2

Sep 2003

258510 Posts

Quote:
 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.

Quote:
 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.

Quote:
 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.

Quote:
 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.

Quote:
 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 2p-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.

Quote:
 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).

2016-09-26, 13:16   #9
GP2

Sep 2003

258510 Posts

Quote:
 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 Mp 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.

2016-09-26, 13:34   #10
GP2

Sep 2003

A1916 Posts

Quote:
 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?

2016-09-26, 15:03   #11
xilman
Bamboozled!

"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across

132·67 Posts

Quote:
 Originally Posted by GP2 Isn't it both?
Not obviously so. Not obvious to me, anyway, but I may be quite wrong.

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