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You have (X+1)^2k = X^2k + 1 + 2(...) so ( X+1 )^2k = 0 mod ( X^2k +1 , 2 )

In other words (X^n+1) is nilpotent in any ring F2[X]/(X^n+1) where n is even, so it remains to concentrate on odd values of n

try something like for i in range(2,10): A=R.quo(x^(2*i+1)+1);j=2 while A((x+1)^j)<>A(x+1): j=j+1

You have (X+1)^2k = X^2k + 1 + 2(...) so ( X+1 )^2k = 0 mod ( X^2k +1 , 2 )

In other words (X^n+1) is nilpotent in any ring F2[X]/(X^n+1) where n is even, so it remains to concentrate on odd values of n

try something like like this (N rather small...)

for i in range(2,10): range(2,10):

A=R.quo(x^(2*i+1)+1);j=2

while A((x+1)^j)<>A(x+1):

A=R.quo(x^(2*i+1)+1);j=2
while A((x+1)^j)<>A(x+1):
j=j+1j=j+1


You have (X+1)^2k = X^2k + 1 + 2(...) so ( X+1 )^2k = 0 mod ( X^2k +1 , 2 )

In other words (X^n+1) is nilpotent in any ring F2[X]/(X^n+1) where n is even, so it remains to concentrate on odd values of n

try something like this (N rather small...)

for i in range(2,10):

A=R.quo(x^(2*i+1)+1);j=2

while A((x+1)^j)<>A(x+1):

j=j+1

while A((x+1)^j)<>A(x+1): j=j+1

You have (X+1)^2k = X^2k + 1 + 2(...) so ( X+1 )^2k = 0 mod ( X^2k +1 , 2 )

In other words (X^n+1) is nilpotent in any ring F2[X]/(X^n+1) where when n is even, so it remains to concentrate on odd values of n

try something like this (N rather small...)

for i in range(2,10):

A=R.quo(x^(2*i+1)+1);j=2 while A((x+1)^j)<>A(x+1): j=j+1

You have (X+1)^2k = X^2k + 1 + 2(...) so ( X+1 )^2k = 0 mod ( X^2k +1 , 2 )

In other words (X^n+1) (X+1)^n is nilpotent in any ring F2[X]/(X^n+1) when n is even, so it remains to concentrate on odd values of n

try something like this (N rather small...)

for i in range(2,10):

A=R.quo(x^(2*i+1)+1);j=2 while A((x+1)^j)<>A(x+1): j=j+1