Difference between revisions of "Dirichlet's theorem"
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'''Dirichlet's theorem''' states: | '''Dirichlet's theorem''' states: | ||
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| − | ''There are infinitely many [[prime number]]s in | + | ''There are infinitely many [[prime number]]s in an [[arithmetic progression]] of the form an + b if a and b are relatively prime (i.e., (a, b) = 1)).'' |
</blockquote> | </blockquote> | ||
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| + | For example, the sequence of numbers which are one more than a multiple of 11 (1, 12, 23, 34, 45, ...) is an arithmetic progression. Dirichlet's theorem states that this sequence contains infinitely many prime numbers. Equivalently, there are infinitely many values of k for which 11*k+1 is a prime number. | ||
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| + | Gauss proposed this conjecture but could not prove it. Dirichlet first proved this in 1837 using the Dirichlet L-series. The proof is quite challenging to understand, and is beyond the scope of most number theory courses. | ||
[[category:mathematics]] | [[category:mathematics]] | ||
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Revision as of 04:24, October 1, 2009
Dirichlet's theorem states:
There are infinitely many prime numbers in an arithmetic progression of the form an + b if a and b are relatively prime (i.e., (a, b) = 1)).
For example, the sequence of numbers which are one more than a multiple of 11 (1, 12, 23, 34, 45, ...) is an arithmetic progression. Dirichlet's theorem states that this sequence contains infinitely many prime numbers. Equivalently, there are infinitely many values of k for which 11*k+1 is a prime number.
Gauss proposed this conjecture but could not prove it. Dirichlet first proved this in 1837 using the Dirichlet L-series. The proof is quite challenging to understand, and is beyond the scope of most number theory courses.