1 | initial version |

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

2 | No.2 Revision |

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+1~~j=j+1

3 | No.3 Revision |

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~~

4 | No.4 Revision |

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

5 | No.5 Revision |

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

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