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 Register FAQ Search Today's Posts Mark Forums Read 2017-04-25, 03:07 #1 devarajkandadai   May 2004 22×79 Posts 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.   2017-04-25, 08:56 #2 Nick   Dec 2012 The Netherlands 1,453 Posts 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).   2017-04-26, 05:44   #3

May 2004

31610 Posts Conjecture pertaining to Gaussian integers

Quote:
 Originally Posted by Nick 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?   2017-04-26, 11:33   #4
Nick

Dec 2012
The Netherlands

1,453 Posts Quote:
 Originally Posted by devarajkandadai So we can take it as proved?
Yes, it follows directly from the course material on this forum.   2017-04-27, 04:59 #5 devarajkandadai   May 2004 22·79 Posts 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.   2017-04-27, 08:44 #6 Brian-E   "Brian" Jul 2007 The Netherlands 326410 Posts @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.   Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post Stargate38 Aliquot Sequences 40 2019-11-30 11:14 devarajkandadai Number Theory Discussion Group 12 2017-12-25 05:43 devarajkandadai Number Theory Discussion Group 11 2017-10-28 20:58 Nick Number Theory Discussion Group 8 2016-12-07 01:16 Sam Kennedy Programming 3 2012-12-16 08:38

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