Conjecture pertaining to Gaussian integers

[devarajkandadai]

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.

[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).

[devarajkandadai]

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?

[Nick]

Originally Posted by devarajkandadai

So we can take it as proved?

Yes, it follows directly from the course material on this forum.

[devarajkandadai]

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.

[Brian-E]

@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.