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 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 5·353 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

22·79 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

5·353 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 2·3·5·109 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.

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