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For a finite integer field $\mathbb{Z}_n$ with a generator $g$ and $x\in\mathbb{Z}_n$, a property I do not understand arises and I have not come across an explanation for it. If another set of values $x_1=n+x,\ x_2=2n+x\dots$ then $g^{x}\neq g^{x_1}\neq g^{x_2}\dots$. Why is this? I've noticed and conjectured that for an $x_k=kn+x$, $g^{x}=g^{x_k +k}$ at least for several examples I have been toying with to get a feel for the mechanics. Since all these values are members of the same congruence class, I'd expect them to generate identical field elements. Is it because the multiplicative subgroup is of order $n-1 $ due to the removal of the additive identity? I'd appreciate any guidance and clarification on this matter.

Ken Goss
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  • Is $n$ a prime number? – Bernard Dec 03 '18 at 23:19
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    Every element of $\mathbb{F}_p$ satisfies $g^p = g$, so $g^{2p} = g^2, g^{3p} = g^3$, etc. In other words, since the multiplicative group has order $p - 1$, the exponents are numbers that make sense $\bmod (p - 1)$, not $\bmod p$. In general exponentiation takes in two different types of objects as input; the fact that exponentiation seems like it takes in the same types of objects as input over the real numbers is misleading. – Qiaochu Yuan Dec 03 '18 at 23:22
  • @Bernard yes it's a prime modulus, sorry for not including that detail explicitly. – Ken Goss Dec 03 '18 at 23:38
  • @Qiaochu Yuan, could you elaborate on the meaning of the phrase "exponentiation takes in two different types of objects..." please? – Ken Goss Dec 03 '18 at 23:40
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    I have a long rant about this; I wrote a (bad) version of it here: https://math.stackexchange.com/questions/56663/is-there-a-natural-way-to-extend-repeated-exponentiation-beyond-integers – Qiaochu Yuan Dec 03 '18 at 23:45
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    @QiaochuYuan you’re right that is a long rant, but I rather like it. – Randall Dec 04 '18 at 00:47

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If your $n$ is a prime number, by lil' Fermat, we have $g^n\equiv g\mod n$, hence $$g^{x_k}=g^{x+k}\enspace\forall k\in\mathbf Z.$$

Bernard
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  • Ah, so straight-forward. I just didn't make the connection. Thank you! Between your answer and Qiaochu Yuan's comments it is clear to me now. – Ken Goss Dec 03 '18 at 23:53