Forum: Miscellaneous Math
2023-01-30, 00:05
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Replies: 40
Views: 3,796
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...
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Forum: Miscellaneous Math
2022-12-22, 14:36
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Replies: 40
Views: 3,796
[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...
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Forum: Miscellaneous Math
2022-12-20, 04:35
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Replies: 40
Views: 3,796
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....
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Forum: Miscellaneous Math
2022-12-19, 00:06
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Replies: 40
Views: 3,796
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)...
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Forum: Miscellaneous Math
2022-12-18, 17:49
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-12-18, 05:08
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Replies: 40
Views: 3,796
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...
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Forum: Miscellaneous Math
2022-12-14, 02:31
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-12-13, 22:29
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-12-13, 21:59
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-12-13, 21:07
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Replies: 40
Views: 3,796
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...
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Forum: Miscellaneous Math
2022-11-27, 16:47
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Replies: 40
Views: 3,796
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...
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Forum: Miscellaneous Math
2022-11-27, 01:04
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-11-26, 17:05
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Replies: 40
Views: 3,796
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.
...
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Forum: Miscellaneous Math
2022-11-26, 10:17
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Replies: 40
Views: 3,796
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...
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Forum: Miscellaneous Math
2022-11-21, 06:51
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Replies: 40
Views: 3,796
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...
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Forum: Miscellaneous Math
2022-11-20, 05:34
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-11-15, 07:30
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-11-14, 11:47
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Replies: 40
Views: 3,796
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...
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Forum: Miscellaneous Math
2022-11-11, 09:37
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-11-11, 02:12
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-11-10, 23:41
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-11-10, 21:47
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Replies: 40
Views: 3,796
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),...
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Forum: Miscellaneous Math
2022-11-10, 21:09
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Replies: 40
Views: 3,796
-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...
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Forum: Miscellaneous Math
2022-11-10, 04:27
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Replies: 40
Views: 3,796
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Forum: Miscellaneous Math
2022-11-10, 02:08
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Replies: 40
Views: 3,796
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