Integer ($\mathbb{Z}$)

An integer is a whole number, positive, negative, or zero.

For an integer $a$ and positive integer $b$, Euclidean division writes $a=bq+r$ with unique integer quotient $q$ and remainder $0\le r<b$. For $a\ge 0$, repeated quotient-and-remainder steps yield the digits of a base-p expansion when $b=p$. The uniqueness follows because the difference of two permitted remainders cannot be a nonzero multiple of $b$.

A finite integer interval is a set $\{a,a+1,\dots ,b\}$ with integers $a\le b$. Its length is $b-a+1$. Infinite integer intervals can also be defined by allowing one or both endpoints to be infinite; a bounded-gaps condition on words always quantifies over finite integer intervals.

In positional notation of base $b$, a digit string $d_n{d}_{n-1}\cdots d_0$ represents ${\sum }_{j=0}^n{d}_j{b}^j$.

A hereditary base representation expands not only the number in base $b$, but also every exponent recursively in that same base. Changing the base at every depth defines the Goodstein base-change operation; replacing it by omega defines an ordinal rank of a Goodstein term.

Start at base two. At each nonzero step, change every base occurrence in the hereditary base representation to the next base and subtract one. Although the natural-number terms can grow, their ordinal ranks of a Goodstein term decrease strictly. Keep zero fixed once it is reached.

Every Goodstein sequence reaches zero. Before zero, the ordinal rank of a Goodstein term of each step is strictly smaller than the preceding one. An infinite sequence would contradict well-foundedness of the ordinals below epsilon zero.

Replacing every base occurrence by omega in a hereditary base representation gives a Cantor normal form below epsilon zero. Hereditary base change preserves this ordinal expression, and subtraction of one lowers it strictly. The ordinal is a termination measure even when the natural-number value increases.

The Goodstein function records the least step at which the Goodstein sequence from a given initial number reaches zero. With the initial step numbered zero, its values on $0,1,2,3$ are $0,1,3,5$. Goodstein's theorem makes this effective termination-time function total.

For an integer base $p\ge 2$, a nonnegative integer has a finite positional representation with digits $k_i\in \{0,\dots ,p-1\}$. Repeated Euclidean division gives the digits uniquely; appending zero high-place digits has no effect. The base need not be prime for uniqueness, but a prime base is useful in Lucas theorem and modular binomial coefficients.

The binary numeral system is positional notation of base two, using the digits zero and one.

The decimal numeral system is positional notation in base ten.

A decimal representation is a positional notation of base ten.

A positive integer is an integer greater than zero.

The parity of an integer records whether it is divisible by two.

An odd integer is not divisible by two; equivalently, it has the form $2k+1$.

An integer is even when it is divisible by two.

An integer is odd when it is not divisible by two.

A square-free integer is divisible by no square greater than one.

A positive integer is squarefull when every prime dividing it also divides it to at least the second power. Thus $n$ is not squarefull as soon as some prime $p$ satisfies $p\mid n$ but $p^2\nmid n$.

A square number is an integer of the form $n^2$ for some integer $n$. Consecutive nonnegative squares differ by $(n+1{)}^2-n^2=2n+1$.