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Primes of the form $ ax+b$

Next we turn to primes of the form $ ax+b$. We assume that $ \gcd(a,b)=1$, because otherwise there is no hope that $ ax+b$ is prime infinitely often. For example, $ 3x+6$ is only prime for one value of $ x$.

Proposition 2.1   There are infinitely many primes of the form $ 4x-1$.

Why might this be true? Let's list numbers of the form $ 4x-1$ and underline the ones that are prime:

$\displaystyle \underline{3}, \underline{7}, \underline{11}, 15, \underline{19},...
...derline{23}, 27, \underline{31}, 35, 39,
\underline{43}, \underline{47}, \ldots$

It certainly looks plausible that underlined numbers will continue to appear. The following PARI program can be used to further convince you:
   f(n, s=0) = for(x=1, n, if(isprime(4*x-1), s++); s

Proof. The proof is similar to the proof of Euclid's Theorem, but, for variety, I will explain it in a slightly different way.

Suppose $ p_1, p_2,\ldots, p_n$ are primes of the form $ 4x-1$. Consider the number

$\displaystyle N = 4p_1\times p_2 \times \cdots \times p_n - 1.$

Then $ p_i \nmid N$ for any $ i$. Moreover, not every prime $ p\mid N$ is of the form $ 4x+1$; if they all were, then $ N$ would also be of the form $ 4x+1$, which it is not. Thus there is a $ p\mid N$ that is of the form $ 4x-1$. Since $ p\not= p_i$ for any $ i$, we have found another prime of the form $ 4x-1$. We can repeat this process indefinitely, so the set of primes of the form $ 4x-1$ is infinite. $ \qedsymbol$

Example 2.2   Set $ p_1=3$, $ p_2=7$. Then

$\displaystyle N = 4\times 3 \times 7 - 1 = \underline{83}
$

is a prime of the form $ 4x-1$. Next

$\displaystyle N = 4\times 3 \times 7\times 83 - 1 = \underline{6971},
$

which is a again a prime of the form $ 4x-1$. Again:

$\displaystyle N = 4\times 3 \times 7\times 83\times 6971 - 1 = 48601811 = 61 \times \underline{796751}.
$

This time $ 61$ is a prime, but it is of the form $ 4x+1 = 4\times 15+1$. However, $ 796751$ is prime and $ (796751-(-1))/4 = 199188$. We are unstoppable

$\displaystyle N = 4\times 3 \times 7\times 83\times 6971 \times 796751 - 1 = \underline{5591}\times 6926049421.
$

This time the small prime, $ 5591$, is of the form $ 4x-1$ and the large one is of the form $ 4x+1$. Etc!

Theorem 2.3 (Dirichlet)   Let $ a$ and $ b$ be integers with $ \gcd(a,b)=1$. Then there are infinitely many primes of the form $ ax+b$.

The proof is out of the scope of this course. You will probably see a proof if you take Math 129 from Cornut next semester.


next up previous
Next: How many primes are Up: Lecture 4: The Sequence Previous: There are infinitely many
William A Stein 2001-09-19