--- In

primenumbers@yahoogroups.com,

Kermit Rose <kermit@...> wrote:

> m**2 A**4 - m * (m-2) A**2 B**2 + B**4

> = ( m * A**2 - m * A * B + B**2 )

> * ( m * A**2 + m * A * B + B**2)

That 4th degree equation is useful (in the

Aurifeullian sense) only at m=2. As I explained,

the general Aurifeullian identity for Phi(n,x)

requires (among other things) that x = m*s^2,

where m divides n. For example:

Phi(n,b)=subst(polcyclo(n),x,b);

print(factor(Phi(4,2*x^2))[,1]~)

[2*x^2 - 2*x + 1, 2*x^2 + 2*x + 1]

print(factor(Phi(6,3*x^2))[,1]~)

[3*x^2 - 3*x + 1, 3*x^2 + 3*x + 1]

print(factor(Phi(5,5*x^2))[,1]~)

[25*x^4 - 25*x^3 + 15*x^2 - 5*x + 1,

25*x^4 + 25*x^3 + 15*x^2 + 5*x + 1]

print(factor(Phi(15,5*x^2))[,1]~)

[625*x^8 - 625*x^7 + 250*x^6 - 125*x^5 + 75*x^4 - 25*x^3 + 10*x^2 - 5*x + 1,

625*x^8 + 625*x^7 + 250*x^6 + 125*x^5 + 75*x^4 + 25*x^3 + 10*x^2 + 5*x + 1]

print(factor(Phi(30,3*x^2))[,1]~)

[81*x^8 - 81*x^7 + 54*x^6 - 27*x^5 + 9*x^4 - 9*x^3 + 6*x^2 - 3*x + 1,

81*x^8 + 81*x^7 + 54*x^6 + 27*x^5 + 9*x^4 + 9*x^3 + 6*x^2 + 3*x + 1]

David