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# ${\mathbb{Z}}_{n}$

Let $n\in\mathbb{Z}$. An equivalence relation, called congruence, can be defined on $\mathbb{Z}$ by $a\equiv b\operatorname{mod}n$ iff $n$ divides $b-a$. Note first of all that $a\equiv b\operatorname{mod}n$ iff $a\equiv b\operatorname{mod}|n|$. Thus, without loss of generality, only nonnegative $n$ need be considered. Secondly, note that the case $n=0$ is not very interesting. If $a\equiv b\operatorname{mod}0$, then $0$ divides $b-a$, which occurs exactly when $a=b$. In this case, the set of all equivalence classes can be identified with $\mathbb{Z}$. Thus, only positive $n$ need be considered. The set of all equivalence classes of $\mathbb{Z}$ under the given equivalence relation is called ${\mathbb{Z}}_{n}$.

Some mathematicians consider the notation ${\mathbb{Z}}_{n}$ to be archaic and somewhat confusing. This matter of notation is most considerable when $n=p$ for some prime $p$, as ${\mathbb{Z}}_{p}$ is used to refer to the $p$-adic integers. To avoid this confusion, some mathematicians use the notation $\mathbb{Z}/n\mathbb{Z}$ instead of ${\mathbb{Z}}_{n}$. On the other hand, the notation ${\mathbb{Z}}_{n}$ should not cause confusion when $n$ is not prime, and is an intuitive shorthand way to write $\mathbb{Z}/n\mathbb{Z}$. Thus, others use ${\mathbb{F}}_{p}$ when $n=p$ for some prime $p$ and ${\mathbb{Z}}_{n}$ otherwise. (The explanation of the usage of $\mathbb{F}_{p}$ will come later.) Still others, especially those who are unfamiliar with the $p$-adic integers, use the notation ${\mathbb{Z}}_{n}$ exclusively. (In this entry, the notation ${\mathbb{Z}}_{n}$ is used exclusively, though it is highly recommended to use another notation when $n=p$ for some prime $p$.)

One usually identifies an element of ${\mathbb{Z}}_{n}$ (which is technically a class, not a number) with the unique element $r$ in the class such that $0\leq r<n$. One can use the division algorithm to establish that, for each class, an $r$ as described exists uniquely. (The set of all $r$’s as described is an example of a residue system.) Thus, the sets ${\mathbb{Z}}_{n}$ are finite with exactly $n$ elements. Addition and multiplication operations can also be defined on ${\mathbb{Z}}_{n}$ in a natural way that corresponds to the operations on $\mathbb{Z}$. Under these operations, ${\mathbb{Z}}_{n}$ is a commutative ring with identity as well as a cyclic ring with behavior $1$. When $n=p$ for some prime $p$, ${\mathbb{Z}}_{n}$ is a field. In this case, the notation ${\mathbb{F}}_{p}$ highlights the fact that the structure is a field. When $n$ is composite, ${\mathbb{Z}}_{n}$ has zero divisors and thus is neither a field nor an integral domain. Also note that ${\mathbb{Z}}_{1}$ is a zero ring, since all integers are equivalent, yielding only one equivalence class.

The $n$ in both ${\mathbb{Z}}_{n}$ and $a\equiv b\operatorname{mod}n$ is called the *modulus*. Performing computations such as addition, subtraction, multiplication, and taking exponents in one of the rings ${\mathbb{Z}}_{n}$ is called *modular arithmetic*.

## Mathematics Subject Classification

13-00*no label found*13M05

*no label found*11-00

*no label found*

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## Comments

## Congruences on Markov numbers

PrimeFan, I'll transfer MarkovNumber to you. I can't access Acta Arithmetica PDFs from where I'm at right now, so I trust that you have read it and will make the appropriate additions.