A first look at Dirichlet’s theorem

As a result of Euclid’s proof that there are infinitely many prime numbers (dated back to the time around 300 BC), we know that there are infinitely many prime numbers dispersed among the odd integers 1, 3, 5, 7, 9, \cdots. It turns out that when you divide the odd integers into 2 halves (see the following two sequences), each half also contains infinitely many prime numbers:

    1, 5, 9, 13, 17, 21, 25 \cdots

    3, 7, 11, 15, 19, 23, 27 \cdots

So does each of the following sequences:

    5, 11, 17, 23, 29, 35, 41 \cdots

    7, 37, 67, 97, 127, 157, 187 \cdots

These examples provides a first look at the Dirichlet’s theorem on arithmetic progressions.
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The Dirichlet’s Theorem on Arithmetic Progressions

An arithmetic progression is a sequence of numbers where each term is obtained by adding a constant to the previous term. The constant is called the common difference. If the first term is a and the common difference is d>0, the following is an arithmetic progression:

    \displaystyle a, a+d, a+2d, a+3d, \cdots

The Dirichlet’s theorem states that an arithmetic progression contains infinitely many prime numbers if the first term a and the common difference d are relatively prime (i.e. the only common divisor of a and d is 1).

The above 4 sequences of integers are all arithmetic progressions where the first term and the common difference are relatively prime. Take the third example 5, 11, 17, 23, 29, 35, 41 \cdots. The common difference is 6 and the first term is 5, which are relatively prime. By the Dirichlet’s theorem, we can be confident that this sequence contains infinitely many prime numbers. Note that not all the terms are prime (e.g. 35). So the prime numbers in an arithmetic progression do not need to be consecutive. In fact, even if an arithmetic progression contains infinitely many primes, there will always be infinitely many terms that are not prime.

Note that in the third example, the first 5 terms are primes (5, 11, 17, 23, 29) but the sixth term is not a prime. Thus 5, 11, 17, 23, 29 is a finite arithmetic progression of length 5 consisting entirely of primes. The first 6 terms 7, 37, 67, 97, 127, 157 in the fourth example is an arithmetic progression of length 6 consisting of entirely of primes.

It is a well known recent result that the prime numbers contains finite arithmetic progressions of any length. This result is known as the Green-Tao theorem and was proved by Ben Green and Terence Tao in 2004.

For a more detailed discussion of the Dirichlet’s theorem, see the Wikipedia entry on Dirichlet’s theorem on arithmetic progressions.

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\copyright \ 2013 \text{ by Dan Ma}

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