20060605, 18:46  #1 
Nov 2003
2^{2}×5×373 Posts 
Check my arithmetic
I derived (by hand) the following polynomial for N = 2,1526L/2,218L =
2^654 + 2^600 + 2^545  2^436  2^382  2^327  2^273  2^218  2^109 + 2^55 + 1 The polynomial is: f(x) = x^6 + 2x^5  10x^4  20x^3 + 16x^2  48x + 72 with root 2^55 + 2^54. This polynomial sends (2z + 1/z) to (64z^12 + 64z^11 + 32z^10  16z^8 16z^7  8z^6  8z^5  4z^4 + 2z^2 + 2z + 1)/z^6 The 12'th degree polynomial is equal to 2,1526L/2,218L with z = 2^54 (or should be if I did the arithmetic correctly) Would someone with access to Maple/Mathematica please check this? It was tedious to do by hand. i.e. please verify that f(2z + 1/z) equals the 12th degree polynomial divided by z^6. 
20060605, 20:25  #2  
"Robert Gerbicz"
Oct 2005
Hungary
7×199 Posts 
Quote:
(22:20) gp > f(x)=x^6+2*x^510*x^420*x^3+16*x^248*x+72 (22:20) gp > substpol(f(x),x,2*z+1/z) %7 = (64*z^12 + 64*z^11 + 32*z^10  16*z^8  176*z^7 + 56*z^6  8*z^5  4*z^4 + 2*z^2 + 2*z + 1)/z^6 (22:20) gp > So there are errors in your polynom. Last fiddled with by R. Gerbicz on 20060605 at 20:27 

20060605, 21:19  #3 
"Robert Gerbicz"
Oct 2005
Hungary
7·199 Posts 
There was also an error in N:
N=2,1526L/2,218L=2^654+2^600+2^5452^4362^3822^3272^2732^218+2^109+2^55+1 I think the correct polynom ( I've found this by hand ) is the following: (checking this by PariGp ): (23:15) gp > f(x)=x^6+2*x^510*x^420*x^3+16*x^2+32*x+8 (23:15) gp > g(z)=substpol(f(x),x,2*z+1/z) %4 = (64*z^12 + 64*z^11 + 32*z^10  16*z^8  16*z^7  8*z^6  8*z^5  4*z^4 + 2*z^2 + 2*z + 1)/z^6 (23:15) gp > Further checking to see that g(z) has a root of 2^54 modulo N: (23:28) gp > lift(g(Mod(2^54,N))) %13 = 0 (23:28) gp > Last fiddled with by R. Gerbicz on 20060605 at 21:30 
20060605, 23:49  #4  
Nov 2003
2^{2}×5×373 Posts 
Quote:
well Thanks. 

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