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What exactly does $\mathbb{Z}_{7429}$ mean?
Is it the set of all integers up to and including 7429?

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up vote 8 down vote accepted

I choose to answer with a more complete explanation because I once also thought that $\mathbb{Z}_n = \{ -(n-1), ..., 0, ..., n-1\}$

To fully understand this group, you must understand the equivalence relation congruence modulo $n$. We say that $(a,b) \in R$ or using an equivalent notation $aRb$ iff $a = b+nk$ $,k \in \mathbb{Z}$.

As an example: congruence modulo $2$.

$(2,4)$ is an element of this relation because $2 = 4 + 2k$, namely $k=-1$.

Now we define the equivalence class of the relation congruence modulo $n$ as following: $$\bar{a} = \{b \mid aRb \}$$ going back to our example of congruence modulo $2$, this would mean that, for example: $$\bar{2}= \{b\mid2Rb\} = \{b\mid2=b+2k \} = \{0,2,-2,...\} = \bar{4} = ...$$

Now we can finally define $\mathbb{Z}_n$: $$\mathbb{Z}_n = \{\bar{0},...,\overline{n-1}\}$$

Hopefully this makes things a bit clearer for you.

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Thanks Jeroen, that makes a bit more sense! – Arvin Nov 5 '11 at 14:17

For the Additive Group of Integers Modulo $m$. which is a cyclic group, the notation $\mathbb{Z}_m$ is often used.

This is the group where $7429\equiv0$ with the usual addition.

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$\mathbb Z_p$ is also used for the ring of $p$-adic integers but this does not apply to 7429 because it is not prime. – lhf Nov 5 '11 at 12:58

Another notation for $\mathbb{Z}/7429\mathbb{Z}.$

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I find this notation better than $\mathbb Z_{7429}$ because of the $p$-adic integers. – user18119 Nov 5 '11 at 13:26
@QiL: Using $\mathbb{Z_{7429}}$ makes me confused as well. – HbCwiRoJDp Nov 5 '11 at 13:29

$\mathbb{Z}_{7429}$ is the set of all integers modulo $7429$.

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