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• ## Re: On the conjecture of Sophie Germain numbers and Fermat Theorem

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• ... To be more precise and more rigorous, this is true for q composite and squarefree. In that case, the set of residues of i^p (mod q) (1
Jan 26, 2005 1 of 7
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> What I have in mind is that for Sophie Germain numbers
> the set of residues of i^p (mod q) (1<i<q) are always comprised
> between 2 and q-1.

To be more precise and more rigorous, this is true for q composite
and squarefree. In that case, the set of residues of i^p (mod q)
(1<i<q) is exactly the set {2,...,q-1}. For q composite and not
squarefree, the set of residues is {0} + {i: 1<i<q and GCD(i,q)=1} +
{some of the multiples of the prime factors of q}. This last set can
be defined more precisely and I have an algorithm to define which
multiples of the prime factors of q are residues. But the main point
is that 1 is never in these sets.
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