Generalized arithmetic progression
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In mathematics, a multiple arithmetic progression, generalized arithmetic progression, or k-dimensional arithmetic progression, is a set of integers 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
- 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
- L(C;P)
be the set of all elements x in Nn of the form
,
with c0 in C, 
in P, and 
in N. L is said to be a linear set if C consists of exactly one element, and P is finite.
A subset of Nn is said to be semilinear if it is a finite union of linear sets.
