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 Showing results 1 to 25 of 26 Search took 0.01 seconds. Search: Posts Made By: paulunderwood
 Forum: Miscellaneous Math 2023-01-30, 00:05 Replies: 40 Views: 3,796 Posted By paulunderwood The test x^(n+1)==-3 (mod n, x^2-3^r*x-3) and... The test x^(n+1)==-3 (mod n, x^2-3^r*x-3) and gcd(r-1,n-1) pans out for n < 10^13 and all r. The verification with GMP+primesieve took several weeks on a dual core Celeron. If the code was run on...
 Forum: Miscellaneous Math 2022-12-22, 14:36 Replies: 40 Views: 3,796 Posted By paulunderwood [452295831401, 5, -2, 9716, 2429, 3644] ... [452295831401, 5, -2, 9716, 2429, 3644] [452295831401, 5, -2, 9716, 2429, 8502] Here z is even for b=-2 and there is no pseudoprime "r=(z+1)/2", yet gcd(2*r-1,n-1) != 1. Another pattern...
 Forum: Miscellaneous Math 2022-12-20, 04:35 Replies: 40 Views: 3,796 Posted By paulunderwood The case for b=-2 I'm back by popular demand! The key with x^2-3^r*x-3 was to transform it to z^2-((-3)^(2*r-1)-2)*z+1 and note that (-3)^(2*r-1)-2 = -3*(3^(2*(r-1))+1)+3-2. So we want to "avoid" 3^(4*(r-1))-1....
 Forum: Miscellaneous Math 2022-12-19, 00:06 Replies: 40 Views: 3,796 Posted By paulunderwood I don't blame you for that. You mentioned... I don't blame you for that. You mentioned earlier we have Fermat PRP and strong Fermat. Both these are no foolproof and PSPs (pseudoprimes) can be constructed. e.g. Carmichael (absolute PSP)...
 Forum: Miscellaneous Math 2022-12-18, 17:49 Replies: 40 Views: 3,796 Posted By paulunderwood I aim to please. Now consider b=3 and b=-3.... I aim to please. Now consider b=3 and b=-3. There are corresponding pseudoprimes: {forstep(n=3,10000000,2, if(!ispseudoprime(n), forstep(b=-3,3,6,...
 Forum: Miscellaneous Math 2022-12-18, 05:08 Replies: 40 Views: 3,796 Posted By paulunderwood The case for 3 Consider working over x^2-3^r*x+3 (**) We want gcd(r-1,n-1)=1 as stated earlier. We also know the multiplicative order z (of 3 mod n) is even and 3^(z/2)==-1 mod n. Now x^2-3^(z/2)*3+3=0 is...
 Forum: Miscellaneous Math 2022-12-14, 02:31 Replies: 40 Views: 3,796 Posted By paulunderwood Instead of hand-waving and saying it just can't... Instead of hand-waving and saying it just can't be true because mathematicians would find the truth of my argument unlikely, please pin point the error in the argument I gave in post #24. Here is...
 Forum: Miscellaneous Math 2022-12-13, 22:29 Replies: 40 Views: 3,796 Posted By paulunderwood Even a large tree starts out as a small one. A... Even a large tree starts out as a small one. A pseudoprime test that has no known counterexample for all parameters for n<2*10^12 (on a Celeron core); A test that may well turn out to be a...
 Forum: Miscellaneous Math 2022-12-13, 21:59 Replies: 40 Views: 3,796 Posted By paulunderwood Pseudoprimes for now, but hopefully leading to a... Pseudoprimes for now, but hopefully leading to a prime test. If the argument just given is valid, then it is one step in the right direction.
 Forum: Miscellaneous Math 2022-12-13, 21:07 Replies: 40 Views: 3,796 Posted By paulunderwood Another attempted proof Having read the paper attached to this thread earlier, rather than considering (-3)^(4*(r-1))-1 if we consider (-3)^(2*(r-1))+1 -- the predecessor -- we have (-3)^(2*(r-1)) == -1 mod n. But then...
 Forum: Miscellaneous Math 2022-11-27, 16:47 Replies: 40 Views: 3,796 Posted By paulunderwood As with "-12" (GCD-less probably because the... As with "-12" (GCD-less probably because the discriminant of at r=znorder(Mod(-12,n)) is 1+48, a square) compared to "12", and comparing "-3" to "3" -- the latter needing GCD depending on whether n...
 Forum: Miscellaneous Math 2022-11-27, 01:04 Replies: 40 Views: 3,796 Posted By paulunderwood Hah, I made a complete blunder. The conclusion is... Hah, I made a complete blunder. The conclusion is 1=1. I have marked it with a red X.
 Forum: Miscellaneous Math 2022-11-26, 17:05 Replies: 40 Views: 3,796 Posted By paulunderwood Partial proof For a basis we know prime $$p$$ that $$x^{p+1} \equiv -3 \pmod{n, x^2-3^rx-3}$$ where the Jacobi symbol of the discriminant is $$-1$$. Now suppose $$n=pq$$ where $$q>1$$ not necessarily prime. ...
 Forum: Miscellaneous Math 2022-11-26, 10:17 Replies: 40 Views: 3,796 Posted By paulunderwood partial results The GCD-less test for "-12" was verrified using Par/GP my forumite mart_t and me up to 1.2*10^12. The test for "-3" has now reached 10^12 using GMP+primesieve. It will take a couple of Celeron CPU...
 Forum: Miscellaneous Math 2022-11-21, 06:51 Replies: 40 Views: 3,796 Posted By paulunderwood I fixed a few things in the paper that were... I fixed a few things in the paper that were mistakes. A GMP+primesieve program is now running here. It is much faster tha the Pari/GP one. I posted on Math.StackExchange but the thread was...
 Forum: Miscellaneous Math 2022-11-20, 05:34 Replies: 40 Views: 3,796 Posted By paulunderwood Here is a short paper about this Lucas(n,3^r,-3)... Here is a short paper about this Lucas(n,3^r,-3) test. Now I have to get into coding up a verification program written with GMP+primesieve. Pari/GP is too slow for the purpose. As I say in the...
 Forum: Miscellaneous Math 2022-11-15, 07:30 Replies: 40 Views: 3,796 Posted By paulunderwood (Lucas) Q=-3 seems the natural base to study as... (Lucas) Q=-3 seems the natural base to study as it is the discriminant of z^2+-z+1. In practice, avoidance of z^2-z+1 seems sufficient. This divides z^3+1 which divides z^6-1. We have...
 Forum: Miscellaneous Math 2022-11-14, 11:47 Replies: 40 Views: 3,796 Posted By paulunderwood A proof? We are busy verifying to 10^12, the test for the test over x^2-12^r*x-12 where x^(n+1)==-12. I notice that the discriminant is 1+48 == 49 for r=0, i.e a square. Anyway back to the "proof" and...
 Forum: Miscellaneous Math 2022-11-11, 09:37 Replies: 40 Views: 3,796 Posted By paulunderwood The pattern broke for +12, with n=20935371731 and... The pattern broke for +12, with n=20935371731 and r=6163, yet gcd(2*r-1,n-1)==145. Testing of -12 is going great guns. Now at n>32,000,000,000, with no output except way points.
 Forum: Miscellaneous Math 2022-11-11, 02:12 Replies: 40 Views: 3,796 Posted By paulunderwood As well as "-12" I am running "+12" overnight.... As well as "-12" I am running "+12" overnight. The latter has the following pattern so far: g := g(2*r-1,n-1). one solution for r up to z the multiplicative order of 12 mod n g = z n%4 =...
 Forum: Miscellaneous Math 2022-11-10, 23:41 Replies: 40 Views: 3,796 Posted By paulunderwood In the "wood" it looks like I found a clearing... In the "wood" it looks like I found a clearing where there is no tree! I could speed up testing by letting a=lift(Mod(-12,n)^r) and pretesting with kronecker(a^2+48,n)==-1 :ermm: :xmastree:
 Forum: Miscellaneous Math 2022-11-10, 21:47 Replies: 40 Views: 3,796 Posted By paulunderwood Indeed! Here is the code I am running: ... Indeed! Here is the code I am running: {b=12;forstep(n=3,1000000000001,2, if(n%1000000000==1,print(n)); if(gcd(b,n)==1&&!ispseudoprime(n)&&Mod(-b,n)^((n-1)/2)==kronecker(-b,n),...
 Forum: Miscellaneous Math 2022-11-10, 21:09 Replies: 40 Views: 3,796 Posted By paulunderwood -12 I tried other bases -2,(-3),-4,-6 and -8. but boy oh boy -12 is absolutely great. The test over x^2-12^r*x-12 has no pseudoprimes less than 300,000,000 and consequently no need to take a GCD. Again I...
 Forum: Miscellaneous Math 2022-11-10, 04:27 Replies: 40 Views: 3,796 Posted By paulunderwood And the pattern is broken with 28027505969 which... And the pattern is broken with 28027505969 which has 4 r's, each with gcd(r-1,n-1)!=1 :ermm:
 Forum: Miscellaneous Math 2022-11-10, 02:08 Replies: 40 Views: 3,796 Posted By paulunderwood The results are consistent so far up to... The results are consistent so far up to 21451730441. The two pseudoprime indices r1 and r2 are given by: r1=z/4+1 r2=3*z/4+1 resulting in r1+r2=2 Rearranging:
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