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August 22nd, 2018, 06:42 PM  #1 
Member Joined: Jul 2010 Posts: 81 Thanks: 1  A most curious primality test...
Long story, but when I came up with this test it was just supposed to be a placeholder of sorts for the actual test I was to be working on. Here it is, written in Javascript (I know, lousy choice for number theory): Code: /* Constraint: N = 1 mod 12 (must be odd N > 3) */ function maybe_prime(N) { var value = big_integer(N); /* Start with base equal to (N  1) / 2 */ var base = value.shiftRight(1); var previous = value.minus(1); for(;;) { if(base.lesser(2)) break; /* Fermat test */ if(!base.modPow(previous, value).equals(1)) return false; /* Divide base in half, discarding any remainder obviously */ base = base.shiftRight(1); } return true; } That said I know there are much better (well I mean PROVEN!) tests such as MillerRabin which perform in more than acceptable running time, but I'd just like to know if there is any theoretical basis at all for this? Most likely just a fluke that will eventually produce a counterexample but... Last edited by Sebastian Garth; August 22nd, 2018 at 06:47 PM. 
August 22nd, 2018, 06:44 PM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,634 Thanks: 567 Math Focus: Yet to find out. 
Java and Javascript are different things btw...

August 22nd, 2018, 06:59 PM  #3 
Member Joined: Jul 2010 Posts: 81 Thanks: 1  
August 22nd, 2018, 09:11 PM  #4 
Senior Member Joined: Aug 2012 Posts: 1,999 Thanks: 572  
August 23rd, 2018, 09:15 AM  #5 
Member Joined: Jul 2010 Posts: 81 Thanks: 1  Oh I just meant the whole backstory of what lead up to me choosing N = 1 mod 12 and so forth. Not really important at this point. But the more I think about it the less faith I have in this test anyhow. I just can't see any number theoretic basis whatsoever for what's going on here. Eventually the law of large numbers will probably prove this to be false. Interesting anomaly though. 
August 23rd, 2018, 09:10 PM  #6 
Newbie Joined: Aug 2018 From: Sydney Posts: 6 Thanks: 1 
You're doing the Fermat test repeatedly with different values. On each occasion, the chance of getting 1 by chance if N isn't prime is about 1/N. For largish N the probablity that you'll get 1 by chance is about (1/N) ^ (log2(N)  2), so the function has an exceedingly low probability of wrongly concluding that N is prime.

August 24th, 2018, 12:08 AM  #7  
Member Joined: Jul 2010 Posts: 81 Thanks: 1  Quote:
 
August 24th, 2018, 02:00 AM  #8 
Newbie Joined: Aug 2018 From: Sydney Posts: 6 Thanks: 1  For smaller numbers you're not doing many tests, which increases the chances of a false positive. Requiring that the number be congruent to 1 mod 12 excludes the possibility that it's a multiple of 2 or 3, which eliminates a large number of candidates that are definitely not prime whatever the primality test might say.

August 24th, 2018, 02:51 AM  #9  
Member Joined: Jul 2010 Posts: 81 Thanks: 1  Quote:
 

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