modified Euler's generalisation of Fermat's theorem

devarajkandadai · 2017-07-07, 05:36

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.

Nick · 2017-07-07, 13:56

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.

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.

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2017-07-07, 13:56	#2
Nick Dec 2012 The Netherlands 2×7×131 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.