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 Showing results 1 to 25 of 35 Search took 0.01 seconds. Search: Posts Made By: paulunderwood
 Forum: Miscellaneous Math 2022-02-05, 06:26 Replies: 34 Views: 6,449 Posted By paulunderwood Generalization II I have now devised a general test of the polynomial x^2-b^r*x+b with an Euler PRP test for the discriminant b^(2*r)-4*b: {tst(n,b,r)=local(t=lift(Mod(b,n)^(2*r-1)),k=kronecker(b,n));...
 Forum: Miscellaneous Math 2022-01-31, 15:32 Replies: 34 Views: 6,449 Posted By paulunderwood Generalization Based on a Lucas PRP test over x^2-b^r*x+b, I have come up with a general test: {tst(n,b,r)=local(t=lift(Mod(b,n)^(2*r-1))); kronecker(b,n)==-1&& kronecker(t-4,n)==1&& gcd(t-3,n)==1&&...
 Forum: Miscellaneous Math 2021-10-26, 05:03 Replies: 34 Views: 6,449 Posted By paulunderwood version for bash The attached program returns exit(0) iff the input is 2-Euler PRP and Lucas PRP. (It also uses gcd((r-1)*(2*r-1),n-1)<=2.) // gcc -o prp_v2 prp_v2.c -lgmp // bash use case:- /* for i in...
 Forum: Miscellaneous Math 2021-10-24, 15:38 Replies: 34 Views: 6,449 Posted By paulunderwood 2 selfidge version We can fuse the Euler and Lucas test thusly: { tst(n,r)=local(fr=lift(Mod(4,n)^r)); gcd((r-1)*(2*r-1),n-1)<=2&& kronecker(fr-8,n)==-1&& Mod(Mod(2*x,n),x^2-(fr/2-2)*x+1)^((n+1)/2)==2; }
 Forum: Miscellaneous Math 2021-10-24, 08:25 Replies: 34 Views: 6,449 Posted By paulunderwood A slight speed up Rather than taking gcd(4^r-4,n)==1 and gcd(4^r-2,n)==1 the substitute gcd((r-1)*(2*r-1),n-1)<=2 is quicker, resulting in the test: { tst(n,r)=local(k=kronecker(2,n);fr=lift(Mod(4,n)^r));...
 Forum: Miscellaneous Math 2021-10-20, 02:59 Replies: 34 Views: 6,449 Posted By paulunderwood GMP code for test #3 I have coded up test #3 from the above paper. // gcc -o prp prp.c -lgmp // usages:- // ./prp // ./prp // echo "print()" | gp -q | ./prp
 Forum: Miscellaneous Math 2021-07-08, 03:01 Replies: 34 Views: 6,449 Posted By paulunderwood Revised paper Here us the revised paper. I'll leave the original up for contrast, :book: :book:
 Forum: Miscellaneous Math 2021-07-07, 19:40 Replies: 34 Views: 6,449 Posted By paulunderwood It occurred to me that since the tests involve... It occurred to me that since the tests involve t^2+something that only half of t might be used. For example: { tst(n)=local(t=2,k=kronecker(-2,n),limit=2*log(n)*log(log(n)),l=0,nm1d2=(n-1)/2);...
 Forum: Miscellaneous Math 2021-07-06, 15:13 Replies: 34 Views: 6,449 Posted By paulunderwood Four Lucas Tests Here my paper distilled from this thread :book:
 Forum: Miscellaneous Math 2021-07-02, 01:44 Replies: 34 Views: 6,449 Posted By paulunderwood If r=(z+2)/4 then the quadratic x^2+(t^2/2+2)*x+1... If r=(z+2)/4 then the quadratic x^2+(t^2/2+2)*x+1 is x^2+(2^((z+2)/2)/2+2)*x+1 is x^2+(2^(z/2)*2/2+2)*x+1. Since 2^(z/2)==-1 then the quadratic reduces to x^2+x+1 which is cyclotomic. A similar...
 Forum: Miscellaneous Math 2021-06-29, 01:55 Replies: 34 Views: 6,449 Posted By paulunderwood For my blanket testing of all r, I notice those n... For my blanket testing of all r, I notice those n that require a gcd are 5 mod 6. This will speed up a little of my search where "the pattern" holds. Status: all 2-PSPs < 3*10^10 and using "the...
 Forum: Miscellaneous Math 2021-06-28, 15:58 Replies: 34 Views: 6,449 Posted By paulunderwood Algorithm Here is the algorithm for x^2-2^r*x-2 { tst(n)=local(t=2); \\ t=2^r if(n==2||n==3,return(1)); \\ trivialiies if(n%2==0||issquare(n)||Mod(-2,n)^((n-1)/2)!=kronecker(-2,n),return(0)); \\ even...
 Forum: Miscellaneous Math 2021-06-28, 11:58 Replies: 34 Views: 6,449 Posted By paulunderwood Assuming z%4==2 and ... Assuming z%4==2 and Mod(Mod(x,n),x^2+(t^2/2+2)*x+1)^((n+1)/2)==kronecker(-2,n) are necessary conditions, -- poof anyone? -- the next to break "the pattern" is { for(v=305962,#V,n=V[v];...
 Forum: Miscellaneous Math 2021-06-27, 21:38 Replies: 34 Views: 6,449 Posted By paulunderwood Only x^2-2^r*x-2 Just when I was about to give up on this thread I have a great test based on x^2-2^r*x-2, which can be transformed into Mod(-2,n)^((n-1)/2)==kronecker(-2,n)...
 Forum: Miscellaneous Math 2021-06-26, 20:32 Replies: 34 Views: 6,449 Posted By paulunderwood Fibonacci Back to the drawing board... Restrict n to 3 or 7 mod 20. Then x^(n+1) == -b^2 (mod n, x^2-b*x-b^2) can be transformed into x^((n+1)/2) == -1 (mod n, x^2+3*x+1) and b^(n-1) == 1 (mod n). I am now...
 Forum: Miscellaneous Math 2021-06-25, 21:25 Replies: 34 Views: 6,449 Posted By paulunderwood I am slightly change the last test for practical... I am slightly change the last test for practical purposes { tst(n,r)=local(t=lift(Mod(2,n)^r)); if(n==2||n==3||n==5,return(1)); if(gcd(30,n)!=1||issquare(n),return(0));...
 Forum: Miscellaneous Math 2021-06-25, 12:39 Replies: 34 Views: 6,449 Posted By paulunderwood I have tested n < 3*10^11. If r=1 the this... I have tested n < 3*10^11. If r=1 the this test is the same as the \$620 quest given on this page (https://www.d.umn.edu/~jgreene/baillie/Baillie-PSW.html) which purports to have around 740...
 Forum: Miscellaneous Math 2021-06-23, 18:29 Replies: 34 Views: 6,449 Posted By paulunderwood GMP is an order quicker than Pari/GP for this... GMP is an order quicker than Pari/GP for this task. I have attached my GMP code. Note that it misses out x^2-x-4, but it would take a minute to test this special case with Pari/GP. I fixed that...
 Forum: Miscellaneous Math 2021-06-21, 23:34 Replies: 34 Views: 6,449 Posted By paulunderwood Only x^2-2^r*x-4 Based on only x^2-2^r*x-4 ... And it's clunky: { tst(n,r)=local(t=lift(Mod(2,n)^r)); ((n%4==1&&gcd(t^2+4,n)==1)|| (n%4==3&&gcd(t^2+8,n)==1))&& kronecker(t^2+16,n)==-1&&
 Forum: Miscellaneous Math 2021-06-21, 20:49 Replies: 34 Views: 6,449 Posted By paulunderwood base 3 variant This time based on x^2-3^r*x+-9 ... n%4==1: { tst(n,r)=local(t=lift(Mod(3,n)^r)); gcd(t^2-3,n)==1&&gcd(t^2-9,n)==1&& kronecker(t^2-4*9,n)==-1&& Mod(3,n)^(n-1)==1&&
 Forum: Miscellaneous Math 2021-06-21, 16:48 Replies: 34 Views: 6,449 Posted By paulunderwood 1+2 selfridges Based on x^2-2^r*x+-4 ... If n%4==1 then: { tst(n,r)=local(t=lift(Mod(2,n)^r)); gcd(t^2-4,n)==1&& kronecker(t^2-16,n)==-1&& Mod(2,n)^(n-1)==1&&
 Forum: Miscellaneous Math 2021-06-20, 17:33 Replies: 34 Views: 6,449 Posted By paulunderwood I have verified this for all r for all n<10^11. ... I have verified this for all r for all n<10^11. I will convert to GMP soon.
 Forum: Miscellaneous Math 2021-06-20, 05:04 Replies: 34 Views: 6,449 Posted By paulunderwood Efficent Test The efficient test is: { tst(n,r)=local(t=lift(Mod(1/2,n)^r)); gcd(t^2-1,n)==1&& gcd(t-2,n)==1&& kronecker(t^2-4*t,n)==-1&& Mod(2,n)^((n-1)/2)==kronecker(t,n)&&...
 Forum: Miscellaneous Math 2021-06-20, 00:22 Replies: 34 Views: 6,449 Posted By paulunderwood Extra gcd This test looks good: { tst(n,r)=local(t=lift(Mod(2,n)^r)); gcd(t^2-1,n)==1&& gcd(2*t-1,n)==1&& kronecker(1-4*t,n)==-1&& Mod(2,n)^((n-1)/2)==kronecker(t,n)&&...
 Forum: Miscellaneous Math 2021-06-19, 08:02 Replies: 34 Views: 6,449 Posted By paulunderwood Restricted domain variation Here I test over x^2-x+2^r where kronecker(1-4*2^r,n)==-1 x^2-x-2^r where kronecker(1+4*2^r,n)==-1 If r is even this is the same as: 2^(n-1)==1 (mod n) x^((n+1)/2)==1 (mod n,...
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