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 2012-11-09, 20:00 #2003 LaurV Romulan Interpreter     Jun 2011 Thailand 52·7·53 Posts what are you talking there? Almost any multiple of 31 would do it (look to the 3rd column after zero) Code: (02:56:07) gp > forstep(i=1,1000,2, a=2^4*31*i;s=sigma(a)-a;v=factorint(s); if(v[1,2]!=4, print(i", "i%31", "i\31",\t"a" , "s", "v))) 31, 0, 1, 15376, 15407, [7, 1; 31, 1; 71, 1] 93, 0, 3, 46128, 77004, [2, 2; 3, 3; 23, 1; 31, 1] 155, 0, 5, 76880, 107818, [2, 1; 31, 1; 37, 1; 47, 1] 217, 0, 7, 107632, 138632, [2, 3; 13, 1; 31, 1; 43, 1] 279, 0, 9, 138384, 261795, [3, 1; 5, 1; 31, 1; 563, 1] 341, 0, 11, 169136, 200260, [2, 2; 5, 1; 17, 1; 19, 1; 31, 1] 403, 0, 13, 199888, 231074, [2, 1; 31, 1; 3727, 1] 465, 0, 15, 230640, 508152, [2, 3; 3, 1; 31, 1; 683, 1] 527, 0, 17, 261392, 292702, [2, 1; 31, 1; 4721, 1] 589, 0, 19, 292144, 323516, [2, 2; 31, 1; 2609, 1] 713, 0, 23, 353648, 385144, [2, 3; 31, 1; 1553, 1] 775, 0, 25, 384400, 569873, [31, 2; 593, 1] 837, 0, 27, 415152, 816168, [2, 3; 3, 1; 31, 1; 1097, 1] 899, 0, 29, 445904, 477586, [2, 1; 31, 1; 7703, 1] (in fact i said "the easiest way" as a joke, there is no other way, and I was expected the question "what is the harder way?" from some nitpicker and I could reply "raising 31 at 4th power", hehe) Last fiddled with by LaurV on 2012-11-09 at 20:07
2012-11-09, 20:37   #2004
RobertS

Aug 2009
somewhere

197 Posts

Quote:
 Originally Posted by BigBrother Now at line 5078, I'm already sieving the c133.
found by ECM:
prp44 = 23684003947956653714782183866686101195981013 (curve 1 stg2 B1=46000000 sigma=3481469303)

2012-11-09, 21:34   #2005
BigBrother

Feb 2005
The Netherlands

2·109 Posts

Quote:
 Originally Posted by RobertS found by ECM: prp44 = 23684003947956653714782183866686101195981013 (curve 1 stg2 B1=46000000 sigma=3481469303)
Meh, I did 1000 curves @ 11M...

2012-11-09, 22:17   #2006
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3·29·83 Posts

Quote:
 Originally Posted by Dubslow Or any even power. Then the sequence is effectively class 4, which means it needs to the rest (besides 2^4*31^(2n)) needs to be a product of 4 primes or less, but that's not sufficient. (There are some restrictions on what each prime has to be mod 4 or mod 8 etc. Also, and an other prime raised to an even power doesn't count against that 4.)
Quote:
 Originally Posted by LaurV what are you talking there? Almost any multiple of 31 would do it (look to the 3rd column after zero) Code: (02:56:07) gp > forstep(i=1,1000,2, a=2^4*31*i;s=sigma(a)-a;v=factorint(s); if(v[1,2]!=4, print(i", "i%31", "i\31",\t"a" , "s", "v))) 31, 0, 1, 15376, 15407, [7, 1; 31, 1; 71, 1] 93, 0, 3, 46128, 77004, [2, 2; 3, 3; 23, 1; 31, 1] 155, 0, 5, 76880, 107818, [2, 1; 31, 1; 37, 1; 47, 1] 217, 0, 7, 107632, 138632, [2, 3; 13, 1; 31, 1; 43, 1] 279, 0, 9, 138384, 261795, [3, 1; 5, 1; 31, 1; 563, 1] 341, 0, 11, 169136, 200260, [2, 2; 5, 1; 17, 1; 19, 1; 31, 1] 403, 0, 13, 199888, 231074, [2, 1; 31, 1; 3727, 1] 465, 0, 15, 230640, 508152, [2, 3; 3, 1; 31, 1; 683, 1] 527, 0, 17, 261392, 292702, [2, 1; 31, 1; 4721, 1] 589, 0, 19, 292144, 323516, [2, 2; 31, 1; 2609, 1] 713, 0, 23, 353648, 385144, [2, 3; 31, 1; 1553, 1] 775, 0, 25, 384400, 569873, [31, 2; 593, 1] 837, 0, 27, 415152, 816168, [2, 3; 3, 1; 31, 1; 1097, 1] 899, 0, 29, 445904, 477586, [2, 1; 31, 1; 7703, 1] (in fact i said "the easiest way" as a joke, there is no other way, and I was expected the question "what is the harder way?" from some nitpicker and I could reply "raising 31 at 4th power", hehe)
As far as I can tell, every example there has 31^2, which does not contradict what I said. (None of these small multipliers has more than four odd-powered prime factors.)
Code:
>>> for i in range(1,1000,2):
...     n = 2**4*31*i
...     s = a.aliquot(n)
...     v = a.factor(s)
...     if v[2] != 4:
...             print("{}, {}, {}, \t{}={}, \t{}, {}".format(i, i%31, i//31, n, a.factor(n), s, v))
...
31, 0, 1, 	15376=2^4 * 31^2, 	15407, 7 * 31 * 71
93, 0, 3, 	46128=2^4 * 3 * 31^2, 	77004, 2^2 * 3^3 * 23 * 31
155, 0, 5, 	76880=2^4 * 5 * 31^2, 	107818, 2 * 31 * 37 * 47
217, 0, 7, 	107632=2^4 * 7 * 31^2, 	138632, 2^3 * 13 * 31 * 43
279, 0, 9, 	138384=2^4 * 3^2 * 31^2, 	261795, 3 * 5 * 31 * 563
341, 0, 11, 	169136=2^4 * 11 * 31^2, 	200260, 2^2 * 5 * 17 * 19 * 31
403, 0, 13, 	199888=2^4 * 13 * 31^2, 	231074, 2 * 31 * 3727
465, 0, 15, 	230640=2^4 * 3 * 5 * 31^2, 	508152, 2^3 * 3 * 31 * 683
527, 0, 17, 	261392=2^4 * 17 * 31^2, 	292702, 2 * 31 * 4721
589, 0, 19, 	292144=2^4 * 19 * 31^2, 	323516, 2^2 * 31 * 2609
713, 0, 23, 	353648=2^4 * 23 * 31^2, 	385144, 2^3 * 31 * 1553
775, 0, 25, 	384400=2^4 * 5^2 * 31^2, 	569873, 31^2 * 593
837, 0, 27, 	415152=2^4 * 3^3 * 31^2, 	816168, 2^3 * 3 * 31 * 1097
899, 0, 29, 	445904=2^4 * 29 * 31^2, 	477586, 2 * 31 * 7703

2012-11-12, 04:54   #2007
schickel

"Frank <^>"
Dec 2004
CDP Janesville

41128 Posts

Quote:
 Originally Posted by richs How does a 2^4 * 31 driver break?
Quote:
 Originally Posted by Batalov Because of my avatar, I will reserve 804588.
Most elegant is this way....... (from 804588):
Code:
 1947 .  c114 = 2^4 * 31^2 * 9218757242216532814830455218507779571072152947622474831189587924445638638177013241155356122322458800614403093
1948 .  c114 = 2 * 31 * 1217 * 4679
(2^4 * 31 ran from i1643:c64...)

It's good to see that there is still some life in a lot of these sequences!

2012-11-12, 05:51   #2008
LaurV
Romulan Interpreter

Jun 2011
Thailand

100100001110112 Posts

Quote:
 Originally Posted by Dubslow (None of these small multipliers has more than four odd-powered prime factors.)
Hmmm... That is what I missed in your first post, or at least, sounded very complicate as you said it... So, you say that having 5 or more primes different of 2 or 31 in the list, all at odd powers, will make impossible to kill a 2, or breed a new 2. Now, I did not know this, and after a couple of unsuccessful attempts to get a counterexample with pari/gp, I swore I would take the pencil (but not yet). I took gp and did "select 5 random primes, do their product, times 2^4, times an even random power of 31, factor its sigma minus itself, and if the power of 2 is not 4, then print it; repeat forever". When I use 4 primes, it murders few drivers every second (i.e. printing lines), but with 5 or more primes, it prints nothing after 15 minutes. Now I must take the pencil to understand why... And thanks for teaching me something new.

Last fiddled with by LaurV on 2012-11-12 at 05:53 Reason: forgot the "repeat forever" part

2012-11-12, 07:21   #2009
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3·29·83 Posts

Quote:
 Originally Posted by LaurV Hmmm... That is what I missed in your first post, or at least, sounded very complicate as you said it... So, you say that having 5 or more primes different of 2 or 31 in the list, all at odd powers, will make impossible to kill a 2, or breed a new 2. Now, I did not know this, and after a couple of unsuccessful attempts to get a counterexample with pari/gp, I swore I would take the pencil (but not yet). I took gp and did "select 5 random primes, do their product, times 2^4, times an even random power of 31, factor its sigma minus itself, and if the power of 2 is not 4, then print it; repeat forever". When I use 4 primes, it murders few drivers every second (i.e. printing lines), but with 5 or more primes, it prints nothing after 15 minutes. Now I must take the pencil to understand why... And thanks for teaching me something new.
Don't think I'm any sort of genius. If you figure out why, please tell me.

I was guessing/extrapolating from what Clifford wrote.
Quote:
 Originally Posted by Mr. Stern Let a change in the exponent a be termed a mutation. This occurs only when the 2s count of t is equal to or less than the class of 2^a*v. In the former case, the exponent a increases and in the latter, a is reduced to the 2s count of t. The stability of a guide depends upon its class: the smaller the class, the more stable the guide. For example, a class 2 guide will mutate if t is the product of two primes of the form 4n+1 or is a prime of the form 8n+3 or 4n+1. But a class 1 guide mutates only when t is a prime of the form 4n+1.
Simply extrapolate that a class n driver might be broken if it factors into n or less odd-powered primes.

(From another part of what he said, and some numerology on my part, I'm fairly sure that when v in a perfect-driver is raised to an odd power, the overall class is raised to the power of 2. That is, 2^1 * 3^2 is class 1, 2^2 * 7^2 is class 2, and 2^4 * 31^2 is class 4, and class 4 means what it does as above. (Edit: This can easily be proven. For a perfect driver, v=2^p-1, v prime. Thus 2s_count(v) = pow_of_2(sigma(v)) = pow_of_2(2^p) = p. Thus p + (-1) = p-1 = power of two in the driver, as in the example below. (The only part I don't understand is how you can just add the two separate 2s counts to get the class.))
Quote:
 Originally Posted by Stern, Clifford When the class of a driver is zero or -1, a small 2s count of t is not sufficient in itself to effect a change in the exponent a because the 2s count of t is always greater than zero. Help is required from one of the components of v by having its exponent aquire an even power in order to temporarily raise the driver's class above zero. For example, when the 2^2 · 7 driver takes the form 2^2 · 7^2, its class of -1 temporarily increases by 3 (the 2s count of 7) so a mutation will occur when the 2s count of t is 2 or 1.

Last fiddled with by Dubslow on 2012-11-12 at 07:27 Reason: forgot link

2012-11-12, 20:42   #2010
wblipp

"William"
May 2003
New Haven

3×787 Posts

Quote:
 Originally Posted by LaurV So, you say that having 5 or more primes different of 2 or 31 in the list, all at odd powers, will make impossible to kill a 2, or breed a new 2. Now, I did not know this, and after a couple of unsuccessful attempts to get a counterexample with pari/gp, I swore I would take the pencil (but not yet) ... I must take the pencil to understand why... And thanks for teaching me something new.
The next aliquot value is σ(N)-N.

σ(p^a*q^b*s^c)=σ(p^a)*σ(q^b)*σ(s^c)

σ(p)=p+1, so 2|σ(p)

With 5 or more distinct primes at odd powers,

2^4||N and 2^5|σ(N), so 2^4||σ(N)-N

Of course sometime 4|p+1, so you can get to 2^5 with fewer than five distinct primes

Last fiddled with by wblipp on 2012-11-12 at 20:47 Reason: Fix "five or six" errors

 2012-11-14, 11:52 #2011 yoyo     Oct 2006 Berlin, Germany 607 Posts BTW: Is somebody working on the remaining C137?
 2012-11-14, 12:00 #2012 LaurV Romulan Interpreter     Jun 2011 Thailand 243B16 Posts not me
2012-11-15, 17:27   #2013
BigBrother

Feb 2005
The Netherlands

110110102 Posts

Quote:
 Originally Posted by yoyo BTW: Is somebody working on the remaining C137?
I have, there's now a c157...I've done 4500 curves @ 11M.

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