mersenneforum.org  

Go Back   mersenneforum.org > Great Internet Mersenne Prime Search > Math

Reply
 
Thread Tools
Old 2017-04-17, 18:44   #12
JeppeSN
 
JeppeSN's Avatar
 
"Jeppe"
Jan 2016
Denmark

2·34 Posts
Default

The exponent n has the additional (besides 11) prime factor 5402920651. Cofactor is still composite.


The number 10^n + 7 is so huge you cannot primality test it. But you can search for small factors. Knowing the full factorization of the exponent n is maybe a slight advantage when checking for small divisors of 10^n + 7; I do not now.


It is extremely unlikely that an arbitrary number that huge be prime. The average distance between consecutive primes that huge is 9.1 * 10^1522. That is also the expected number of "random" candidates you need to test at this size before you hit a prime.


/JeppeSN
JeppeSN is offline   Reply With Quote
Old 2017-04-17, 18:53   #13
a1call
 
a1call's Avatar
 
"Rashid Naimi"
Oct 2015
Remote to Here/There

7×277 Posts
Default

I am pretty sure that no commercially available (super)computers that can be very useful for a number of that size.
But if you have a deep enough pocket, there might be options.
Let us know your budget.
a1call is offline   Reply With Quote
Old 2017-04-17, 19:10   #14
firejuggler
 
firejuggler's Avatar
 
Apr 2010
Over the rainbow

22·617 Posts
Default

Well, 10^n+7 are 5mod 6, so they have a chance to be prime
firejuggler is offline   Reply With Quote
Old 2017-04-17, 19:19   #15
a1call
 
a1call's Avatar
 
"Rashid Naimi"
Oct 2015
Remote to Here/There

111100100112 Posts
Default

The probability of any random number of that size being prime is greater than 0. But it is an understatement to say it is astronomically low.
There is nothing special about powers of 10 (5 and 2 to be more precise) + a coprime like 7, compared to any other number when it comes to being prime except that it can be easily represented with decimal based notations.
The only thing we can be sure of is that it won't factor into, 2, 5 or 7.

Last fiddled with by a1call on 2017-04-17 at 19:27
a1call is offline   Reply With Quote
Old 2017-04-17, 20:44   #16
VBCurtis
 
VBCurtis's Avatar
 
"Curtis"
Feb 2005
Riverside, CA

2·7·11·29 Posts
Default

Quote:
Originally Posted by a1call View Post
I am pretty sure that no commercially available (super)computers that can be very useful for a number of that size.
But if you have a deep enough pocket, there might be options.
Let us know your budget.
This is total BS. Perhaps you'd like to explain what your imagination has in mind? For a budget, estimate the total wealth of the planet; in fact, you can imagine every physical resource on the planet is dedicated to your "options". Please, let us know how long you think a calculation would take for primality, as well as what algorithm you would be using.
VBCurtis is offline   Reply With Quote
Old 2017-04-17, 20:53   #17
kladner
 
kladner's Avatar
 
"Kieren"
Jul 2011
In My Own Galaxy!

3×17×197 Posts
Default

Quote:
Originally Posted by VBCurtis View Post
This is total BS. Perhaps you'd like to explain what your imagination has in mind? For a budget, estimate the total wealth of the planet; in fact, you can imagine every physical resource on the planet is dedicated to your "options". Please, let us know how long you think a calculation would take for primality, as well as what algorithm you would be using.
I believe the explanation could be that a1call's intention is wry irony.
kladner is offline   Reply With Quote
Old 2017-04-17, 21:43   #18
a1call
 
a1call's Avatar
 
"Rashid Naimi"
Oct 2015
Remote to Here/There

7·277 Posts
Default

I just happen to think that the technology to perform specialized tasks such as factoring large integers much quicker than PCs is neither out of reach (given the right resources), nor perhaps non existent.
What do you know about optical computers?
a1call is offline   Reply With Quote
Old 2017-04-17, 22:10   #19
a1call
 
a1call's Avatar
 
"Rashid Naimi"
Oct 2015
Remote to Here/There

7·277 Posts
Default

Here is the Wikipedia article for those interested:
https://en.m.wikipedia.org/wiki/Optical_computing
a1call is offline   Reply With Quote
Old 2017-04-17, 23:02   #20
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

202618 Posts
Default

Quote:
Originally Posted by JeppeSN View Post
The exponent n has the additional (besides 11) prime factor 5402920651. Cofactor is still composite.


The number 10^n + 7 is so huge you cannot primality test it. But you can search for small factors. Knowing the full factorization of the exponent n is maybe a slight advantage when checking for small divisors of 10^n + 7; I do not now.


It is extremely unlikely that an arbitrary number that huge be prime. The average distance between consecutive primes that huge is 9.1 * 10^1522. That is also the expected number of "random" candidates you need to test at this size before you hit a prime.


/JeppeSN

the only advantage ( though I did try factoring to 2^30 or thereabouts earlier) I see is it might break the number down into values we can easily mod by the totients for other numbers. since it factors into all odd primes, and all odd numbers above 2 are nontotients according to the OEIS, then the prime divisors of the exponent themselves only help to break the number down to more manageable chunks I think. I still hope there's a form we can put the exponent in that will help in finding factors. At least it's a form ( the whole value 10^n+7) that is 5 mod 6 ( if n is above 0)

10^1+7 =6*2+5
10^2+7 =6*17+5
10^3+7 = 6*167+5
...


if you use it algebraically 6*k+5 the next k is roughly (10*previous k) -3 if that holds up we can give a value to figure out if it fits the form of a possible prime (composites show specific forms for k). but that would brute force it even more probably.

Last fiddled with by science_man_88 on 2017-04-17 at 23:10
science_man_88 is offline   Reply With Quote
Old 2017-04-17, 23:18   #21
Nick
 
Nick's Avatar
 
Dec 2012
The Netherlands

5×13×23 Posts
Default

Quote:
Originally Posted by a1call View Post
I just happen to think...
It is time for you to start thinking a little deeper before posting here.

At what fraction of the speed of light do electrical signals travel in wires?

For a task which would take many thousands of years, would a switch to optical signalling make any practical difference?

Once you have worked that out, you may understand the incredulous reactions.
Nick is online now   Reply With Quote
Old 2017-04-17, 23:31   #22
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

8,369 Posts
Default

Quote:
Originally Posted by science_man_88 View Post
though I did try factoring to 2^30 or thereabouts earlier) .
doh that was the whole thing not the exponent so the whole 10^n+7 has no factors up to 2^30:

Code:
my(a=39639600000000033079722130334105193516374462454515382070790605358914985114125041349652662917631491895468078068144227588488256698643731392656301687065828647039364869335097585180284485610320873304664240820377108832285808799206086668871981475355607012901091630830301025471386040357260999007608336089976844194508766126364538537876281839232550446576248759510420112471055243135957657955673172345352299040688058220310949388025140588819053919947072444591465431690373800860072775388686735031425736023817399933840555739331789612967251075090969235858418789282170029771749917300694674164737016209063843863711544823023486602712537214687396625868342705921270261329804829639431028779358253390671518359245782335428382401587826662256037049288785974197816738339397949057227919285478001984783327820046311610982467747270922924247436321534899106847502480979159775057889513728084684088653655309295401918623883559378101223949718822361892160105855110817069136619252398279854449222626529937148527952365200132318888521336420774065497849818061528283162421435659940456500165398610651670525967581872312272576910353953026794574925570625206748263314588157459477340390340721137942441283493218656963281508435329143235196824346675487925901422428051604366523321204101885544161429043996030433344359907376778035064505458154151505127356930201786304995038041680449884220972543830631822692689381409196162752232881243797552100562355276215679788289778365861726761495203440291101554746940125702944095269599735362222957327158451869004300363876943433675157128680119087);forprime(x=3,2^30,if(lift(Mod(10,x)^lift(Mod(a,x-1))+7)==0,print(x)))
science_man_88 is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Bug with particular large number VolMike YAFU 18 2012-04-09 21:39
Wagstaff number primality test? ixfd64 Math 12 2010-01-05 16:36
newbie question - testing primality of very large numbers NeoGen Software 8 2006-03-20 01:22
is GGNFS checking for SNFS number? Washuu Factoring 10 2005-08-11 05:09
checking smaller number fortega Data 2 2005-06-16 22:48

All times are UTC. The time now is 16:01.

Tue Nov 24 16:01:51 UTC 2020 up 75 days, 13:12, 4 users, load averages: 1.72, 1.74, 1.72

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.