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Consider the positive integers
4 and 21. Since these numbers have no prime factors in common (the only
factor of 4 is 2, the only factors of 21 are 3 and 7), the numbers 4
and 21 are said to be relatively prime. Now consider an arithmetic sequence
that is generated by starting with one of the numbers, say 21, and repeatedly
adding the other number to produce the sequence. In this case we would
have the sequence 21, 25, 29, 33, 37, 41, and so on. We could also start
with the 4 to generate 4, 25, 46, 67, 89, and so on. Such a sequence
is called a prime arithmetic progression. Dirichlet's theorem simply
states that any such sequence contains an infinite number of prime numbers.
Peter Dirichlet first proved
this theorem in 1835. The proof is still generally considered to be
difficult to understand, and has not been improved upon much from its
original form.
The theorem is an integral
part of many other significant proofs in number theory.
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