In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear the fact that an object is large, small, or long with little limitation or restraint, respectively. The use of "arbitrarily" often occurs in the context of real numbers (and its subsets thereof), though its meaning can differ from that of "sufficiently" and "infinitely".

The statement

is a shorthand for:

In the common parlance, the term "arbitrarily long" is often used in the context of sequence of numbers. For example, to say that there are "arbitrarily long arithmetic progressions of prime numbers" does not mean that there exists any infinitely long arithmetic progression of prime numbers (there is not), nor that there exists any particular arithmetic progression of prime numbers that is in some sense "arbitrarily long". Rather, the phrase is used to refer to the fact that no matter how large a number ${\displaystyle n}$ is, there exists some arithmetic progression of prime numbers of length at least ${\displaystyle n}$.

Similar to arbitrarily large, one can also define the phrase "${\displaystyle P(x)}$ holds for arbitrarily small real numbers", as follows:

In other words:

While similar, "arbitrarily large" is not equivalent to "sufficiently large". For instance, while it is true that prime numbers can be arbitrarily large (since there are infinitely many of them due to Euclid's theorem), it is not true that all sufficiently large numbers are prime.

As another example, the statement "${\displaystyle f(x)}$ is non-negative for arbitrarily large ${\displaystyle x}$." could be rewritten as: