Generalized arithmetic progression
Encyclopedia
In mathematics
, a multiple arithmetic progression, generalized arithmetic progression, or k-dimensional arithmetic progression, is a set of integer
s constructed as an arithmetic progression
is, but allowing several possible differences. So, for example, we start at 17 and may add a multiple of 3 or of 5, repeatedly. In algebraic terms we look at integers
where a, b, c and so on are fixed, and m, n and so on are confined to some ranges
and so on, for a finite progression. The number k, that is the number of permissible differences, is called the dimension of the generalized progression.
More generally, let
be the set of all elements
in 
of the form
with
in 
, 
in 
, and 
in 
. 
is said to be a linear set if 
consists of exactly one element, and 
is finite.
A subset of
is said to be semilinear if it is a finite union of linear sets.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a multiple arithmetic progression, generalized arithmetic progression, or k-dimensional arithmetic progression, is a set of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s constructed as an arithmetic progression
Arithmetic progression
In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...
is, but allowing several possible differences. So, for example, we start at 17 and may add a multiple of 3 or of 5, repeatedly. In algebraic terms we look at integers
- a + mb + nc + ...
where a, b, c and so on are fixed, and m, n and so on are confined to some ranges
- 0 ≤ m ≤ M,
and so on, for a finite progression. The number k, that is the number of permissible differences, is called the dimension of the generalized progression.
More generally, let
be the set of all elements
in of the formwith
in , in , and in . is said to be a linear set if consists of exactly one element, and is finite.A subset of
is said to be semilinear if it is a finite union of linear sets.