Let [mathematical expression not reproducible] be a positive integer such that s [greater than or equal to] 1 and r [greater than or equal to] 0 are integers, [p.sub.1], [p.sub.2], ..., [p.sub.s] are distinct odd primes, and [r.sub.i] [greater than or equal to] 1 is an integer for all i = 1,2, ..., s.

In the case of Perception Options, the examining attorney argued that the applicant sought to register more than one mark because the 'X' in its drawing referred to a changing single-digit integer. As such, and perhaps to distinguish from the decision in Dial-A-Mattress, the examiner emphasized that the possible combinations in this case were essentially limitless.

Integer parts of real closed fields are especially interesting as they are known to coincide with models of a certain natural fragment of Peano Arithmetic, namely Open Induction (see [S]).

YouTube, owned by search engine giant Google, admitted it never thought that a video would be viewed greater than the 32-bit integer that it had to upgrade to a 64-bit integer.