In mathematics, the phrase arbitrarily large, arbitrarily small, arbitrarily long is used in such statements as:
- "ƒ(x) is non-negative for arbitrarily large x."
which is shorthand for:
- "For every real number n, ƒ(x) is non-negative for some values of x greater than n."
"Arbitrarily large" should not be confused with the phrase "sufficiently large"; e.g., prime numbers can be arbitrarily large (since there are an infinite number of them) and some sufficiently large numbers are not prime. "Arbitrarily large" does not mean "infinitely large" — for instance, while prime numbers can be arbitrarily large, no infinitely large prime exists because all prime numbers (as well as all other integers) are finite.
Such phrases as "P(x) is true for arbitrarily large x" sometimes are so used primarily for emphasis as in "P(x) is true for all x, no matter how large x is"; then the phrase "arbitrarily large" lacks the meaning indicated above and is logically synonymous with "all."
From the existence of "arbitrarily long arithmetic progressions of prime numbers" cannot be inferred that any infinitely long arithmetic progression of prime numbers exists (none do) nor that any particular arithmetic progression of prime numbers in some sense is "arbitrarily long"; rather that no matter how large a number n is, some arithmetic progression of prime numbers of length at least n exists.
The statement "ƒ(x) is non-negative for arbitrarily large x." could be rewritten as:
- "For every real number n exists a real number x exceeding n that ƒ(x) is non-negative.
Using "sufficiently large" instead yields:
- "Such a real number n such exists that for every real number x exceeding n ƒ(x) is non-negative.
- Infinitely Large vs. Arbitratily Large. Accessed 21 February 2012.
- 4 Arbitrarily Large Data. Accessed 21 February 2012