mersenneforum.org  

Go Back   mersenneforum.org > Great Internet Mersenne Prime Search > Math > Number Theory Discussion Group

Reply
 
Thread Tools
Old 2017-04-25, 03:07   #1
devarajkandadai
 
devarajkandadai's Avatar
 
May 2004

22×79 Posts
Default Conjecture pertaining to Gaussian integers

Let a + ib be a Gaussian integer. Let p be a rational integer prime of shape 4m + 3. Then ((a + ib)^(p^2-1) - 1) == 0 (mod p). This is subject to a + ib and p being co-prime.
devarajkandadai is offline   Reply With Quote
Old 2017-04-25, 08:56   #2
Nick
 
Nick's Avatar
 
Dec 2012
The Netherlands

1,453 Posts
Default

Yes, this follows from what we did on Gaussian integers in the discussion group (assuming I have understood you correctly).

Let \(R=\mathbb{Z}[i]/p\mathbb{Z}[i]\), the set of Gaussian integers modulo \(p\). Then \(R\) has \(p^2\) elements.
As \(p\equiv 3\pmod{4}\), \(p\) remains prime in \(\mathbb{Z}[i]\) (see theorem 62) so \(R\) is a finite integral domain and hence a field, and therefore \(R^*\) has \(p^2-1\) elements.
If \(a+bi\) and \(p\) are coprime then \(\overline{a+bi}\) is a unit in \(R\) so raising it to the power \(p^2-1\) gives \(\bar{1}\) by Lagrange's theorem (theorem 83).
Nick is offline   Reply With Quote
Old 2017-04-26, 05:44   #3
devarajkandadai
 
devarajkandadai's Avatar
 
May 2004

31610 Posts
Default Conjecture pertaining to Gaussian integers

Quote:
Originally Posted by Nick View Post
Yes, this follows from what we did on Gaussian integers in the discussion group (assuming I have understood you correctly).

Let \(R=\mathbb{Z}[i]/p\mathbb{Z}[i]\), the set of Gaussian integers modulo \(p\). Then \(R\) has \(p^2\) elements.
As \(p\equiv 3\pmod{4}\), \(p\) remains prime in \(\mathbb{Z}[i]\) (see theorem 62) so \(R\) is a finite integral domain and hence a field, and therefore \(R^*\) has \(p^2-1\) elements.
If \(a+bi\) and \(p\) are coprime then \(\overline{a+bi}\) is a unit in \(R\) so raising it to the power \(p^2-1\) gives \(\bar{1}\) by Lagrange's theorem (theorem 83).
So we can take it as proved?
devarajkandadai is offline   Reply With Quote
Old 2017-04-26, 11:33   #4
Nick
 
Nick's Avatar
 
Dec 2012
The Netherlands

1,453 Posts
Default

Quote:
Originally Posted by devarajkandadai View Post
So we can take it as proved?
Yes, it follows directly from the course material on this forum.
Nick is offline   Reply With Quote
Old 2017-04-27, 04:59   #5
devarajkandadai
 
devarajkandadai's Avatar
 
May 2004

22·79 Posts
Red face Conjecture pertaining to Gaussian integers

Thank you very much. Would be glad if you wouldlet me have your full name; my id: dkandadai@gmail.com. Incidentally, as founder of maths corner on fb let me invite you to join that group.
devarajkandadai is offline   Reply With Quote
Old 2017-04-27, 08:44   #6
Brian-E
 
Brian-E's Avatar
 
"Brian"
Jul 2007
The Netherlands

326410 Posts
Default

@devarajkandadai
Nick goes strictly by first name only, being the privacy-enthusiast that he is. You won't catch him on Facebook. And he's also too polite to mention that this result would be a simple observation for undergraduate students, so mentioning him by name in that context is not really necessary.
Brian-E is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Gaussian Aliquot Sequences? How to run in Pari/GP? Stargate38 Aliquot Sequences 40 2019-11-30 11:14
Conjecture pertaining to modified Fermat's theorem devarajkandadai Number Theory Discussion Group 12 2017-12-25 05:43
Gaussian integers- use of norms devarajkandadai Number Theory Discussion Group 11 2017-10-28 20:58
Basic Number Theory 10: complex numbers and Gaussian integers Nick Number Theory Discussion Group 8 2016-12-07 01:16
Gaussian Elimination Animation Sam Kennedy Programming 3 2012-12-16 08:38

All times are UTC. The time now is 05:47.

Sun Oct 25 05:47:29 UTC 2020 up 45 days, 2:58, 0 users, load averages: 1.10, 1.49, 1.57

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.