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Def/Sequence of primes
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Definition of Sequence of primes: The sequence of primes is the sequence $(p_i)$, defined recursively as follows:
- For $i = 0$, $p_0 = 2$.
- For $i > 0$, $p_i$ is the smallest prime number which is not in the set $\{ p_0, \ldots, p_{i-1} \}.$
The existence of the "smallest prime number which is not in the set $\{ p_0, \ldots, p_{i-1} \}$" follows from the following two facts:
- Every nonempty set of natural numbers has a smallest element.
- There are infinitely many prime numbers.
Sometimes we say "let $(2,3,5, \ldots)$ be the sequence of prime numbers" to refer to the above sequence of primes.
Logical Connections
This definition logically relies on the following definitions and statements: Def/Sequence, State/Every nonempty subset of N has a smallest element, State/There are infinitely many prime numbers
The following statements and definitions logically rely on the material of this page:
To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Basic number theory
