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 2017-07-07, 05:36 #1 devarajkandadai     May 2004 22·79 Posts modified Euler's generalisation of Fermat's theorem When the base is a rational integer Euler's generalisation holds. When the base is a Gaussian integer the tentative rule is as follows: For every prime factor (of the composite number) with shape 4m+1 use Euler's totient.For every prime factor with shape 4m+3 use (p^2-1).Reduce product of above product by a factor of 2 for every prime of shape 4m+1 and by a factor of 4 for every prime prime of shape 4m+3. Needless to say exponent and base should be coprime.
 2017-07-07, 13:56 #2 Nick     Dec 2012 The Netherlands 1,759 Posts Recall that we define the norm of a Gaussian integer $$w=a+bi$$, written $$N(w)$$, by $$N(w)=a^2+b^2$$. The essence of the problem here is to derive the formula for the number of units in the ring of Gaussian integers modulo $$w$$. A good way to start is to show that, as long as $$w\neq 0$$, this ring contains precisely $$N(w)$$ distinct elements.

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