20200923, 22:57  #100  
Jan 2020
2^{3}·3·5 Posts 
Quote:


20200923, 23:16  #101 
Sep 2002
Database er0rr
2×1,723 Posts 
b=2, b=4 and b=8 seems to give counterexamples 7 mod 8, as you can see by running the Pari/GP program given in that post.
Last fiddled with by paulunderwood on 20200923 at 23:17 
20200923, 23:25  #102  
Jan 2020
120_{10} Posts 
Quote:
Since the thread title is "Amazing 6", If b = 6, will the results be linked to b = 3 and/or b = 9? Maybe I should go with the prime numbers such as b = 5 and b = 7. Last fiddled with by tuckerkao on 20200923 at 23:32 

20200923, 23:41  #103  
Sep 2002
Database er0rr
2×1,723 Posts 
Quote:
By plugging in primes to the aforementioned program will show you that sometimes a prime b will give counterexamples readily and sometimes another b not. If a prime does give counterexamples then so will its powers. Likewise powers of b that don't readily give counterexamples seem not to give counterexamples. So maybe the powers are consistent. Although this thread is labelled "Amazing 6" it has rambled on to other tests, all quadratric, some based on quadratic equations and most recently x^22*x+2^r as can be seen in this paper hot off the press. Last fiddled with by paulunderwood on 20200924 at 00:09 

20200928, 03:49  #104  
Sep 2002
Database er0rr
2·1,723 Posts 
Quote:
Last fiddled with by paulunderwood on 20200928 at 03:53 

20200928, 17:04  #105 
Sep 2002
Database er0rr
110101110110_{2} Posts 
x^2y test
I have devised another PRP for odd nonsquare n:
Let y = 2^r1 Then seek gcd(r1,n1)==1 kronecker(y,n)==1 and test Mod(2,n)^(n1)==1 Mod(y,n)^((n1)/2)==1 (x+1)^(n+1) == y + 1 (mod n, x^2  y) Early results are good. Edit: After 32 hours of pure Pari/GP, verification reached 10^12. Clearly GMP is required to take it up an exponent or two. Last fiddled with by paulunderwood on 20200930 at 20:32 
20201001, 20:52  #106 
Sep 2002
Database er0rr
110101110110_{2} Posts 
Pari/GP took 32 hours to verify all "r" up to n<10^12. GMP was just over 4 times faster. I could speed things a little by doing a base2 Euler PRP test on the list of 2PSP I am using, but I will continue with Fermat 2PRP tests. I will report back when I reach the next exponent, in a couple of days.
Edit: 10^13 was reached and no counterexamples found. The GMP program and the required ".dat" file are attached  remove ".txt". Last fiddled with by paulunderwood on 20201004 at 17:10 
20201006, 00:38  #107 
Sep 2002
Database er0rr
2·1,723 Posts 
A slight variation for x^2y
Instead of using base x+1 choose to use base x+2. Then the test is, with any y=2^r1, for nonsquare odd n:
find jacobi(y,n)==1 then test 2^(n1) == 1 (mod n) y^((n1)/2)==1 (mod n) (x+2)^(n+1) == y + 4 (mod n, x^2y) I can replace the Fermat 2PRP test with an Euler 2PRP to speed up verification. Here is the (Euler 2PRP version of) the in GMP plus ".dat" file  remove ".txt" The test has been verified up to 10^12 Last fiddled with by paulunderwood on 20201006 at 14:03 
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