20220706, 15:00  #12 
Aug 2020
79*6581e4;3*2539e3
1112_{8} Posts 
Maybe there's some sarcasm that eludes me, but isn't it correct that they have a low weight? If it was just a hint that it's trivially true, then one man's trivia is another man's <opposite of trivia>.

20220707, 19:27  #13  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
5×7×283 Posts 
Quote:
Quote:
1. There is no weight or even a concept of weight for this sequence. Weight is an estimate of yield by testsieving for a range of n's for a given k. It is like saying "John travelled from Newark to Philadelphia with an average speed of 55 mph. Therefore his speed at the specific light intersection in Edison, NJ was 55 mph." No it wasn't. That average speed has no predction value for the specific speed exactly at that light (and that intersection is the only point of interest  because in this thread k is tied to n; there is only one value!). The speed at that intersection could have been 0 mph for a minute if the light was red; it could have been 40 mph if the light was green. It has nothing to do with the quality, brand or model of that John's car. Or with his average speed over that drive. 2. "k is a large prime" has no association with low or high weight. It could have been high, it could have been low. Test them! (only for disproving "k is a large prime means low weight"; it doesn't). Thought this is meaningless anyway because of see #1. 3. The whole concept of "Proth weight" is out of the window here as well, because these are not Proth (because k > 2^n). Enough reasons? 

20220712, 09:54  #14  
Jul 2022
3 Posts 
Thanks for all those links.
There was so much written on this subject, I appreciate that. Quote:
We don't know much about the factors of those numbers. Ken Wilke showed some dependencies. For P_{49*} we have to trial divide all small primes until we find a factor or perform a costly N1 / PRP test, right? Thanks to ATH who has done a great job searching for factors below 6*10^{12}. A small function in Pari/GP to find a factor could look like this: Code:
tdiv(s,b1,b2)={my(a);s;forprime(p=b1,b2,a=Mod(2,p)^s;if(!Mod(2*a*aa+1,p),return(p)))}; I'm running Code:
tdiv(74207281,6*10^12,10^13) There is a small chance that a factor is found. 

20220712, 16:05  #15  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
5·7·283 Posts 
Quote:
Code:
tdiv(s,b1,b2)={my(a);s;forprime(p=b1,b2,a=Mod(2,p)^s;if(a*(a+a1)+1,,return(p)))}; And you can refactor it into plain C or better yet in OpenCL or CUDA; there is a mulmod and powmod2 function that can be borrowed from R.Gerbicz' polysieve; or take polysieve and adapt it to this task. That will be faster still. 

20220712, 16:50  #16 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
5×7×283 Posts 
Or you can run 64threaded LLR (or PFGW) on an appropriate computer and remove all doubts. (You can get one at AWS.)
First check the relative speed of the two programs whichever is running faster, then queue it and wait. You can use the structured input that LLR accepts (PFGW will accept many more ways to write it): ABC $a^$b$a^$c+1 2 148414561 74207280 If PRP test passes then you can open a bottle of champagne and run the N1 for another slot of time. llr4 shows this ETA: Starting Fermat PRP test of 2^1484145612^74207280+1 Using generic reduction AVX512 FFT length 16128K, Pass1=896, Pass2=18K, clm=2, 32 threads, a = 3 Iteration: 30000 / 148414560 [0.00%], ms/iter: 69.168, ETA: 118d 18:58 ... I think with oFFT_Increment=1 one might get FFT length 16M instead of this size (which has 9x in one pass and 7x in the other); it could be faster. Testing PFGW... ... ... slightly slower, same FFT chosen: PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8] Generic modular reduction using generic reduction AVX512 FFT length 16128K, Pass1=896, Pass2=18K, clm=2, 32 threads on A 148414563bit number Using stopwatch per 2500 iters  ms/iter: 73.20 ETA: ~125d 18:00hrs 
20220712, 19:53  #17  
Einyen
Dec 2003
Denmark
6427_{8} Posts 
Quote:
I'm sorry but I started trial factoring again when you started this thread, I often do this when I'm reminded of old "projects", I'm already at 2.4*10^{13}. 

20220716, 13:30  #18 
Jul 2022
3 Posts 
Thank you Serge Batalov and ATH for the support. With your help the function could be clearly Improved:
Code:
tdiv(s,b1,b2)={my(a);s;forprimestep(p=b1,b2,Mod(1,14)Mod(9,14)Mod(11,14),a=Mod(2,p)^s;if(a*(a+a1)+1,,return(p)))}; I tested LLR 4.0.1 to get an ETA also. Code:
./llr64 d t4 q"2^1484145612^74207280+1" 2^742072811 > 2^74207280, so we can only do a PRP test for 2^1484145612^74207280+1. Current FFT beeing too near the limit, next FFT length is forced... Starting Fermat PRP test of 2^1484145612^74207280+1 Using generic reduction FMA3 FFT length 17920K, Pass1=1792, Pass2=10K, clm=1, 4 threads, a = 3 Iteration: 140000 / 148414560 [0.00%], ms/iter: 283.943, ETA: 487d 06:52 This is too much. My computer is to slow for this kind of test. To move on, I would have liked to run ECM, but neither Yafu nor GMPECM worked for me due to memory overflows. The only thing that worked for me was a P1 test with B1 = B2 = 15k in Yafu: Code:
wine yafux64 v v v B1pm1 15000 B2pm1 15000 "pm1(7*(2^1484145612^74207280+1))" YAFU Version 2.08 Built with Microsoft Visual Studio 1922 Using GMPECM 7.0.4, Powered by MPIR 3.0.0 Detected Intel(R) Core(TM) i58250U CPU @ 3.40GHz Detected L1 = 32768 bytes, L2 = 6291456 bytes, CL = 64 bytes Using 1 random witness for RabinMiller PRP checks Cached 664579 primes; max prime is 9999991 >> pm1: starting B1 = 15K, B2 = 15K on C44677236 Using mpz_mod Using lmax = 8 with NTT which takes about 474MB of memory Using B1=115000, B2=15036, polynomial x^1 P = 3, l = 8, s_1 = 4294967298, k = s_2 = 1, m_1 = 2500 Probability of finding a factor of n digits: 20 25 30 35 40 45 50 55 60 65 0.001 3.5e05 8.7e07 1.6e08 3.3e10 7.3e15 0 0 0 0 Step 1 took 65649070ms pm1: Process took 71745.5287 seconds. pm1: found prp1 factor = 7 ***factors found*** P1 = 7 The test took about 20 hours and Yafu needed almost 6 GB of my total 8 GB memory. The probability finding a factor of 15 digits, if one exists, is around 2 %. Better than nothing. 
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