# Coloring the integers

Let $$S$$ be a set of natural numbers. Let $$f(S)$$ be equal to the minimum number $$k$$ such that it's possible to color every integer (all integers $$\ldots, -2, -1, 0, 1, 2, \ldots$$, not only the set $$S$$) with one of $$k$$ colors, such that two integers whose difference is in $$S$$ have different colors. Let $$f(S) = 0$$ if no such minimum exists (if we need infinite colors). Compute $$f(S)$$ for each of the following sets.

1. $$S = \{1, 2, 4, 8, 16, \ldots\}$$ is the set of powers of 2.
2. $$S = \{2, 4, 6, 8, 10, \ldots\}$$ is the set of positive even integers.
3. $$S = \{2, 3, 5, 7, 11, \ldots\}$$ is the set of prime numbers.
4. $$S = \{2, 3, 4, 5\}$$

Concatenate your answers. For example, if the answers are $$7, 2016, 0, 1$$ in that order, enter $$7201601$$.

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