mersenneforum.org A Universally derided "primality test".
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 2020-09-11, 12:35 #1 RMLabrador     "Roman V. Makarchuk" Aug 2020 Ukraine 2·17 Posts A Universally derided "primality test". Hi, Information here. http://romanvm-prime.com/ Regards, Roman V. Makarchuk Last fiddled with by Uncwilly on 2020-09-22 at 17:50
 2020-09-11, 14:47 #2 paulunderwood     Sep 2002 Database er0rr 11×317 Posts Code: [n,u]=[231, 65];Mod([1,1;1,u],n)^n==[u,-1;-1,1] 1 Theorem 1 also works for composites. Testing (n-1)/2 times with different u is infeasible for large n. Theorem 2 is ill-defined. What are a(u) and c(u). Can you give a numerical example how this works? Last fiddled with by paulunderwood on 2020-09-11 at 14:57
 2020-09-11, 15:40 #3 RMLabrador     "Roman V. Makarchuk" Aug 2020 Ukraine 2×17 Posts Theorem 1 also works for composites. Yes.There is a lot of exeption. For some values u. Carmichael numbers do it even for many of u, but there is not a problem. Not exist such exeption that have BOTH type of resudials for some different u, if so, this numbers is prime. Theorem 2 is ill-defined. What are a(u) and c(u). Can you give a numerical example how this works?[/QUOTE] a(u), c(u), b(u) - polynomial of u, result of analytical powering of matrix. p=5 u^4+u^3+4u^2+5u+5 (1) 5-u+2=7-u, substitute u^4-29u^3+319u^2-1580u+2980 (2) (1)-(2)=5(2u-7)(3u^2-21u+85) (2u-7)(3u^2-21u+85) = g(u) 5=p That true ONLY if p is prime. Last fiddled with by RMLabrador on 2020-09-11 at 15:44
2020-09-11, 16:15   #4
paulunderwood

Sep 2002
Database er0rr

11×317 Posts

Quote:
 Originally Posted by RMLabrador Theorem 1 also works for composites. Yes.There is a lot of exeption. For some values u. Carmichael numbers do it even for many of u, but there is not a problem. Not exist such exeption that have BOTH type of resudials for some different u, if so, this numbers is prime.
Code:
[n,u,w]=[1247, 601, 638];Mod([1,1;1,u],n)^(n)==[u,-1;-1,1]&&Mod([1,1;1,w],n)^(n)==[1,1;1,w]
1
Quote:
Quote:
 Theorem 2 is ill-defined. What are a(u) and c(u). Can you give a numerical example how this works?
a(u), c(u), b(u) - polynomial of u, result of analytical powering of matrix.
p=5
u^4+u^3+4u^2+5u+5 (1)
5-u+2=7-u, substitute
u^4-29u^3+319u^2-1580u+2980 (2)

(1)-(2)=5(2u-7)(3u^2-21u+85)
(2u-7)(3u^2-21u+85) = g(u)

5=p
That true ONLY if p is prime.
I will have to think more about what you are writing.

Last fiddled with by paulunderwood on 2020-09-11 at 16:15

 2020-09-11, 16:42 #5 RMLabrador     "Roman V. Makarchuk" Aug 2020 Ukraine 2·17 Posts Thank You for advice! You are right! And I'm right too when talk about symmetry of resudials, ok 1247 - 601 and 638? try 1247-601+2 and 1247-638+2. For prime numbers, such symmetry exist; Sorry for take Your time, I'm just not search for ordinary exeptions, I'm find out such of them that break the symmetry.
 2020-09-11, 16:56 #6 paulunderwood     Sep 2002 Database er0rr 1101100111112 Posts The characteristic equations of the matrices are: x^2 - (1+u)*x + u-1 == 0 and x^2 - (1+w)*x + w-1 == 0 The "residuals" depend on the jacobi symbol of the discriminants of these equations: Jacobi((1+u)^2-4(u-1), n) jacobi((1+w)^2-4(w-1), n) If the Jacobi symbol is 1 then the matrical test is a glorified Fermat PRP test. If the symbol is -1 then it is a lucas-based test. Recieved wisdom says that any Fermat+Lucas test has counterexamples for freely varying parameters (unless maybe the choice of u and w are more restricted -- which is what I am currently studying). Last fiddled with by paulunderwood on 2020-09-11 at 16:57
 2020-09-11, 19:51 #7 RMLabrador     "Roman V. Makarchuk" Aug 2020 Ukraine 2×17 Posts I'm refined condition of my Theorem 2)) If we write this in modulo form, we got Carmichael numbers as exception for some u, in strict form there in no exeption. Modulo form have a flaw... I guess, You see my point? First, about symmetry and modulo flaw? And please check this too http://romanvm-prime.com/?page_id=145 Last fiddled with by RMLabrador on 2020-09-11 at 20:25
 2020-09-12, 12:47 #8 RMLabrador     "Roman V. Makarchuk" Aug 2020 Ukraine 2·17 Posts See the real test on the page, Theorem 4 below on page. United Magic of Symmetry and Complex Numbers) romanvm-prime Last fiddled with by Uncwilly on 2020-09-12 at 17:02 Reason: You have already posted your URL twice. No need to spam.
2020-09-12, 15:01   #9
paulunderwood

Sep 2002
Database er0rr

11×317 Posts

Quote:
 Originally Posted by RMLabrador See the real test on the page, Theorem 4 below on page. United Magic of Symmetry and Complex Numbers) romanvm-prime.com
Code:
n=3225601;A=Mod(Mod([1+x,1;1,0],n),x^2+1);B=Mod(Mod([1,1;1,x],n),x^2+1);R=lift(lift(A^n-B^n));print(R)
[x, 0; 0, 3225600*x]

 2020-09-12, 20:05 #10 Dr Sardonicus     Feb 2017 Nowhere 1110101111112 Posts I did notice one unusual situation: The characteristic polynomial is x^2 - (u + 1)*x + (u - 1); the discriminant is u^2 - 2*u + 5. As long as this is nonzero (mod p), there are two eigenvalues, and it is at least not insane to think about diagonalizing the matrix, which might be probative. If p == 1 (mod 4) however, i2 == -1 (mod p) has two solutions. Taking u = 1 + 2*i makes the discriminant of the characteristic polynomial 0, so it has a repeated factor: x^2 - (2 + 2*i)*x + 2*i = (x - (1 + i))^2. In this case, there is only one eigenvalue, 1 + i, of multiplicity 2. The matrix A - (1+i)I (I = 2x2 identity matrix) is (using Pari-GP syntax) N = [-i,1;1,i] which is nilpotent of index 2. Either column [-i;1] or [1;i] of N serves as an eigenvector of A. Thus A = (1 + i)*I + N. Also, I and N commute, so A^p == (1+i)^p*I + N^p (mod p), and N^p is the 2x2 zero matrix. Thus, A^p == (1+i)^p*I == (1 + i)*I (mod p) in this case. Examples: p = 5, i = 2, u = 1 + 2*2 = 0, 1 + i = 3; or i = 3, u = 1 + 2*3 = 2, 1 + i = 4 p = 13, i = 5, u = 1 = 2*5 = 11, 1 + i = 6; or i = 8, u = 1 + 2*8 = 4, 1 + i = 9 Last fiddled with by Dr Sardonicus on 2020-09-12 at 20:08 Reason: xingif ostpy
 2020-09-13, 15:31 #11 RMLabrador     "Roman V. Makarchuk" Aug 2020 Ukraine 2×17 Posts Very nice! I love this place. I talk that using modulo computation lead to false "exception" and once again do it by themselves)) We must do something with to avoid error like this P*B == 0 mod P situation where part P must be an outcome of the test and give 0. Most of false "exception" arise from this, when B is some polynomial, and B give zero to us for some test condition for non-prime P instead of expected P Can not emphasize this more clearly in my poor English Do You understand me? We can build such test. Symmetry is one of the game changer. The second way exist too. We can not deal with modulo and we can not (at lest now) do the test without modulo)) Once again, go to the bottom of my page

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