mersenneforum.org > Math primility testing
 Register FAQ Search Today's Posts Mark Forums Read

 2003-12-21, 10:57 #1 andi314     Nov 2002 7410 Posts primility testing I hope somebody can help me: I have a number ( approx. 15-20 digits) and i try to trial factor it to p(1000)=7919 and start afterwards a fermat test ( a^p mod p=a) with the bases a=2,3,5,7. Can i then say that the tested number is prime???
 2003-12-21, 12:11 #2 michael   Dec 2003 Belgium 1018 Posts No, you can't, but you have a pretty good assumption. If you want to use Fermat's little theorem as a primality test for a number p you must test all prime exponents smaller than p-1. Pseudoprimes for each base aren't very rare, but combinations of a few bases (4 like you did) gives you a good idea... -michael
 2003-12-21, 12:18 #3 andi314     Nov 2002 2×37 Posts i thought that carmichael numbers are really rare and can be excluded by doing a little trail factoring.
 2003-12-21, 12:27 #4 michael   Dec 2003 Belgium 5×13 Posts They are not very frequent, but there are still enough to not be sure about the primality of a number... If i'm correct there are 3 numbers smaller than 100.000 that pass fermat's little theorem for base 2,3,5 and 7 that are composite: 29341 = 13x37x61 46657 = 13x37x97 75361 = 11x13x17x31 -michael
 2003-12-21, 20:49 #5 andi314     Nov 2002 10010102 Posts but these numbers are eliminated by trial factoring up to 7000!!!
 2003-12-21, 20:54 #6 michael   Dec 2003 Belgium 6510 Posts But these numbers are only 5 digits too, i wouldn't know how big the factors could become for 15-20 digit carmichael numbers... -michael Last fiddled with by michael on 2003-12-21 at 20:55
 2003-12-21, 21:10 #7 michael   Dec 2003 Belgium 6510 Posts 721801 = 601x1201, that's already a little bigger, let's see if i can find some pseudoprimes to base 2,3,5,7 with only factors bigger than 7000... Also: A number n is a pseudoprime to the base b if b^n-1 is congruent to 1 modulo n. A Carmichael number is a composite number n such that b^n-1 is congruent to 1 modulo n for every b that is relatively prime to n. So we are talking about composite numbers that are pseudoprime to bases 2,3,5 and 7 ... this are not necessarily Carmichael numbers. -michael
2003-12-22, 02:45   #8

"Richard B. Woods"
Aug 2002
Wisconsin USA

22·3·641 Posts
Re: primility testing

Quote:
 Originally posted by andi314 Can i then say that the tested number is prime ... i thought that carmichael numbers are really rare and can be excluded by doing a little trail factoring. ... but these numbers are eliminated by trial factoring up to 7000!!!???
There is a big mathematical difference between a statement that can be proved to be true and a statement that is true 99.9999...% of the time but sometimes, even if very rarely, is false..

If one claims, or wants to say, simply that "[a certain number] is prime", that implies that it can be proven, absolutely and without any possible doubt, that the number is 100% definitely a prime number.

If, instead, one is referring to a number which has been proven by trial factoring to have no factors smaller than some limit (e.g., 7000 or even 7 trillion) and, in addition, passes the Fermat pseudoprime test for 4 (or even 4000) different prime bases, then one can correctly say that the number is a probable prime, but one cannot correctly say that the number is a "prime" with no qualifying adjective ahead of the word "prime". So it's always necessary to refer to the latter example as a "probable prime" or "pseudoprime".

 2003-12-22, 09:42 #9 andi314     Nov 2002 10010102 Posts so which other methods could i use to determine that a number is 100% prime???
2003-12-22, 12:49   #10
smh

"Sander"
Oct 2002
52.345322,5.52471

29×41 Posts

Quote:
 Originally posted by andi314 so which other methods could i use to determine that a number is 100% prime???
Try http://www.alpertron.com.ar/ECM.HTM

 2003-12-22, 13:23 #11 andi314     Nov 2002 10010102 Posts but i want to implement this methods in a program so i cant use a webpage!!

 Similar Threads Thread Thread Starter Forum Replies Last Post kladner Soap Box 3 2016-10-14 18:43 pepi37 Software 6 2013-04-12 09:42 grobie Marin's Mersenne-aries 1 2006-05-15 12:26 eepiccolo Math 6 2006-03-28 20:53 ndpowell Math 4 2005-06-26 20:14

All times are UTC. The time now is 22:46.

Tue May 18 22:46:47 UTC 2021 up 40 days, 17:27, 0 users, load averages: 1.42, 1.53, 1.67