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:

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

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

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

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

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

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

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

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

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

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

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

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.