20081124, 10:40  #1 
Sep 2008
7 Posts 
Method of Euclid's Proof
I was wondering why the method of Euclid's proof isn't used to find world's largest primes...
To find a new prime,you could just multiply all primes up to a certain number, and then add one. If you're sure you didn't miss a prime you've found a new one. maybe the primes found this way don't grow fast enough? I'm probably missing a simple thing. 
20081124, 10:50  #2  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
6,329 Posts 
Quote:
After having done all the multiplies and added 1 now how do you know if the result is a prime? You need some sort of test to determine if your number is prime. Not as easy as it might seem. And even if you determine it is not prime, how do you find the divisors? Again you need some sort of method to extract the necessary information. It is not good enough just to say "Oh, there is a new prime in there somewhere, but I don't know what it is". Last fiddled with by retina on 20081124 at 10:51 

20081124, 11:17  #3  
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
2^{2}×11^{2}×23 Posts 
Quote:
2*3*5*7*11*13 + 1 = 30031 59 * 509 = 30031 Paul 

20081124, 13:32  #4 
Feb 2006
Denmark
230_{10} Posts 
This is based on a common misunderstanding of Euclid's proof. As Xilman's example shows, there are actually two possiblities: The number may be prime or it may be composite with prime factors which are larger than the multiplied primes. In the vast majority of tested cases, and probably almost all cases, it is the second possibility. Then there is no known feasible method to find the large prime factors.

20081129, 20:27  #5  
Aug 2006
3·1,993 Posts 
Finding a new prime is easy. I put
Code:
nextprime(random(10^65)) 70486640254630306486542498129083911644039810089312084380333291127 (I then verified this in about twotenths of a second with isprime(%)), a prime that in all likelihood no one has ever seen before. Quote:


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