Number Theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into several fields, according to the methods used and the type of questions investigated. (See the list of number theory topics.)
The term "arithmetic" is also used to refer to number theory. This is a somewhat older term, which is no longer as popular as it once was. Number theory used to be called the higher arithmetic, but this too is dropping out of use. Nevertheless, it still shows up in the names of mathematical fields (arithmetic functions, arithmetic of elliptic curves, arithmetic geometry). This sense of the term arithmetic should not be confused either with elementary arithmetic, or with the branch of logic which studies Peano arithmetic as a formal system. Mathematicians working in the field of number theory are called number theorists.

In number theory, Sylvester's sequence is a sequence of integers in which each member of the sequence is the product of the previous members, plus one. Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880.
Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions with the same sum. The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its members. Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds, and hard instances for online algorithms.

Time-keeping on a clock gives an example of modular arithmetic, the "clock group" is represented by the group Z/12Z for a 12-hour clock and Z/24Z for a 24-hour clock.

                                               Did you know?

• ...that every positive integer can be expressed as the sum of four squares of integers?
• ...that it is impossible to separate any power higher than the second into two like powers?
• ...that the only integer that can be written as reverse powers is 16, i.e. 16 = 2^4 = 4^2?