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
is, there exists some arithmetic progression of prime numbers of length at least
[1] Similar to arbitrarily large, one can also define the phrase "
holds for arbitrarily small real numbers", as follows:[2] 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.
Instead, the usage in this case is in fact logically synonymous with "all".