20171019, 06:15  #1 
May 2004
2^{2}·79 Posts 
Gaussian integers use of norms
Let f(x) = a^x + c = m where a and x belong to N; c is a Gaussian integer.Then a^(x+k*norm(m)) + c = = 0 (mod m). Here k belongs to N.
Last fiddled with by devarajkandadai on 20171019 at 06:16 
20171019, 09:07  #2 
May 2004
2^{2}×79 Posts 
That should read a^(x+ k*Eulerphi(norm(m)) +c = = 0 (mod m).
Last fiddled with by devarajkandadai on 20171019 at 09:08 
20171020, 10:51  #3 
Dec 2012
The Netherlands
2^{5}·3^{2}·5 Posts 
There is a missing bracket in the 2nd post.
Are you claiming that this is true for all natural numbers a,x,k and all Gaussian integers c? There appear to be obvious counterexamples  but perhaps I have not understood you correctly. 
20171020, 14:48  #4  
Feb 2017
Nowhere
23×151 Posts 
Quote:
a^(x+ k*Eulerphi(norm(m))) + c == 0 (mod m) then substituting a = 2, x = 1, c = 2, m = 4 gives 2^(1 +8*k) + 2 == 0 (mod 4) which only holds for k = 0. Exercise: Supply an additional hypothesis, under which your statement becomes correct. 

20171027, 04:24  #5 
May 2004
2^{2}×79 Posts 
Yesthis can be tested if you have pari.

20171027, 05:08  #6  
May 2004
2^{2}·79 Posts 
Quote:


20171027, 06:01  #7 
May 2004
100111100_{2} Posts 
Yes this can be easily tested if you have pari. btw did you attend the AMSBENELUX conference at Antwerp in May 1996? I was there.

20171027, 13:07  #8 
Feb 2017
Nowhere
23×151 Posts 

20171028, 05:25  #9  
May 2004
100111100_{2} Posts 
Quote:


20171028, 14:23  #10  
Feb 2017
Nowhere
23·151 Posts 
Quote:
Quote:
No matter. Your attempt to obviate my counterexample by imposing an ad hoc, post hoc condition, is rendered nugatory by the following, just as easily constructed example. Taking a = 10, x = 1, c = 1 + 2*I, m = 11 + 2*I, norm(m) = 125 we obtain 10^(1 + 125*k) + 1 + 2*I == 0 mod (11 + 2*I) The only integer k for which this holds is k = 0. Now, please go wipe the egg off your face, and consider the exercise I proposed. 

20171028, 15:39  #11 
Dec 2012
The Netherlands
5A0_{16} Posts 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Search for prime GaussianMersenne norms (and GMcofactors)  Cruelty  Proth Prime Search  158  20200731 22:23 
Prime numbers norms of modulo cyclotomic polynomials  carpetpool  carpetpool  24  20171029 23:47 
Conjecture pertaining to Gaussian integers  devarajkandadai  Number Theory Discussion Group  5  20170427 08:44 
Basic Number Theory 10: complex numbers and Gaussian integers  Nick  Number Theory Discussion Group  8  20161207 01:16 
Gaussian Elimination Animation  Sam Kennedy  Programming  3  20121216 08:38 