2017-07-07, 05:36 | #1 |
May 2004
100111100_{2} 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 |
Dec 2012
The Netherlands
1744_{10} 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. |
Thread Tools | |
Similar Threads | ||||
Thread | Thread Starter | Forum | Replies | Last Post |
Conjecture pertaining to modified Fermat's theorem | devarajkandadai | Number Theory Discussion Group | 12 | 2017-12-25 05:43 |
Modified Fermat's theorem | devarajkandadai | Number Theory Discussion Group | 14 | 2017-11-12 20:04 |
Modified Fermat pseudoprime | devarajkandadai | Number Theory Discussion Group | 0 | 2017-06-24 12:11 |
Modified Fermat's theorem | devarajkandadai | Number Theory Discussion Group | 2 | 2017-06-23 04:39 |
Modified fermat's last theorem | Citrix | Math | 24 | 2007-05-17 21:08 |