Generalized arithmetic progression

In mathematics, a multiple arithmetic progression, generalized arithmetic progression, k-dimensional arithmetic progression or a linear set, is a set of integers or tuples 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

where ${\displaystyle a,b,c}$ and so on are fixed, and ${\displaystyle m,n}$ and so on are confined to some ranges

and so on, for a finite progression. The number  ${\displaystyle k}$, that is the number of permissible differences, is called the dimension of the generalized progression.

More generally, let

be the set of all elements ${\displaystyle x}$ in ${\displaystyle N^{n}}$ of the form

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