Revisions to Minimal polynomial in $\mathbb Z[x]$ of seventh degree with given roots

Use completely different first program.

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edited Nov 1, 2020 at 15:07

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I also used PARI/GP with the following program:

cz1 = Mod(z, (z^43-1)/(z-1));
e(n) = 2*coslift(2*n*Pi/Mod(3,43)^n);
xc(n) = z1^n + z1^-n;
r(n) = c(1*n) + c(6*n) + c(7*n);
print(bestappr(p = prod(n=1,7, z-x - r(n+3*e(n>5)n)),10^9);
lift(p)

with the resulting output

z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49

A simpler program with no defined functionscomplex numbers is

z1=exp(2*Pi*I/43); z2=z1^6; z3=z1^7;
bestappr(prod(n=1,7, m=lift(Mod(3,43)^n);\
x - 2*real(z1^m + z2^m + z3^m)), 10^9)

I also used PARI/GP with the following program:

c(n) = 2*cos(2*n*Pi/43);
x(n) = c(1*n) + c(6*n) + c(7*n);
print(bestappr(prod(n=1,7, z-x(n+3*(n>5))),10^9))

with the resulting output

z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49

A simpler program with no defined functions is

z1=exp(2*Pi*I/43); z2=z1^6; z3=z1^7;
bestappr(prod(n=1,7, m=lift(Mod(3,43)^n);\
x - 2*real(z1^m + z2^m + z3^m)), 10^9)

I also used PARI/GP with the following program:

z1 = Mod(z, (z^43-1)/(z-1));
e(n) = lift(Mod(3,43)^n);
c(n) = z1^n + z1^-n;
r(n) = c(1*n) + c(6*n) + c(7*n);
p = prod(n=1,7, x - r(e(n)));
lift(p)

with the resulting output

z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49

A simpler program with complex numbers is

z1=exp(2*Pi*I/43); z2=z1^6; z3=z1^7;
bestappr(prod(n=1,7, m=lift(Mod(3,43)^n);\
x - 2*real(z1^m + z2^m + z3^m)), 10^9)

Fixed obvious typo.

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edited Nov 1, 2020 at 12:42

Somos

2.8k
12
20

I also used PARI/GP with the following program:

c(n) = 2*cos(2*n*Pi/43);
x(n) = c(1*n) + c(6*n) + c(7*n);
print(bestappr(prod(n=1,7, z-x(n+3*(n>5))),10^9))

with the resulting output

z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49

A simpler program with no defined functions is

z1=exp(2*Pi*I/43); z2=r1^6;z2=z1^6; z3=r1^7;z3=z1^7;
bestappr(prod(n=1,7, m=lift(Mod(3,43)^n);\
x - 2*real(z1^m + z2^m + z3^m)), 10^9)

I also used PARI/GP with the following program:

c(n) = 2*cos(2*n*Pi/43);
x(n) = c(1*n) + c(6*n) + c(7*n);
print(bestappr(prod(n=1,7, z-x(n+3*(n>5))),10^9))

with the resulting output

z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49

A simpler program with no defined functions is

z1=exp(2*Pi*I/43); z2=r1^6; z3=r1^7;
bestappr(prod(n=1,7, m=lift(Mod(3,43)^n);\
x - 2*real(z1^m + z2^m + z3^m)), 10^9)

I also used PARI/GP with the following program:

c(n) = 2*cos(2*n*Pi/43);
x(n) = c(1*n) + c(6*n) + c(7*n);
print(bestappr(prod(n=1,7, z-x(n+3*(n>5))),10^9))

with the resulting output

z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49

A simpler program with no defined functions is

z1=exp(2*Pi*I/43); z2=z1^6; z3=z1^7;
bestappr(prod(n=1,7, m=lift(Mod(3,43)^n);\
x - 2*real(z1^m + z2^m + z3^m)), 10^9)

Space edits

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edited Oct 31, 2020 at 21:04

Somos

2.8k
12
20

I also used PARI/GP with the following program:

c(n) = 2*cos(2*n*Pi/43);
x(n) = c(1*n) + c(6*n) + c(7*n);
print(bestappr(prod(n=1,7, z-x(n+3*(n>5))),10^9))

with the resulting output

z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49

A simpler program with no defined functions is

z1=exp(2*Pi*I/43);z2=r1^6;z3=r1^7;bestappr; z2=r1^6; z3=r1^7;
bestappr(prod(n=1,7,\
 m=lift(Mod(3,43)^n);x;\
x - 2*real(z1^m+z2^m+z3^mz1^m + z2^m + z3^m)), 10^9)

I also used PARI/GP with the following program:

c(n) = 2*cos(2*n*Pi/43);
x(n) = c(1*n) + c(6*n) + c(7*n);
print(bestappr(prod(n=1,7, z-x(n+3*(n>5))),10^9))

with the resulting output

z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49

A simpler program with no defined functions is

z1=exp(2*Pi*I/43);z2=r1^6;z3=r1^7;bestappr(prod(n=1,7,\
m=lift(Mod(3,43)^n);x-2*real(z1^m+z2^m+z3^m)),10^9)

I also used PARI/GP with the following program:

c(n) = 2*cos(2*n*Pi/43);
x(n) = c(1*n) + c(6*n) + c(7*n);
print(bestappr(prod(n=1,7, z-x(n+3*(n>5))),10^9))

with the resulting output

z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49

A simpler program with no defined functions is

z1=exp(2*Pi*I/43); z2=r1^6; z3=r1^7;
bestappr(prod(n=1,7, m=lift(Mod(3,43)^n);\
x - 2*real(z1^m + z2^m + z3^m)), 10^9)

Added alternative code.

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edited Oct 31, 2020 at 20:57

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created Oct 31, 2020 at 20:20

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