I need to prove that there are infinitely many primes with remainder of 2 when divided by 3. I started out similarly to Euclid's classic proof of an infinite number of prime numbers:
Suppose there is only a finite set of prime numbers with remainder of 2 when divided by 3, then we can write their product as:
$$ P = q_1 \cdot q_2 \cdots q_r, \qquad \text {for some integer } r, = $$ $$ (3q_1+2)\cdot(3q_2+2)\cdots(3q_r+2), $$ for integers $q_r$.
This is where I am stuck. I do not know how to get to a similar contradiction as Euclid did when he considered $P$+$1$ and how the $q_i$'s could not divide $P$+$1$ since they divided $P$. (If they divided $P$+$1$ then they would divide $P$ and $1$, where dividing $1$ is the contradiction). Any ideas on how I can get to a similar contradiction?
Origin — Elementary Number Theory — Jones — p28 — Exercise 2.6
Best Answer
Let $q_1,q_2,\dots,q_n$ be odd primes of the form $3k+2$. Consider the number $N$, where $$N=3q_1q_2\dots q_n+2.$$ It is clear that none of the $q_i$ divides $N$, and that $3$ does not divide $N$.
Since $N$ is odd and greater than $1$, it is a product of one or more odd primes. We will show that at least one of these primes is of the form $3k+2$.
The prime divisors of $N$ cannot be all of the shape $3k+1$. For the product of any number of (not necessarily distinct) primes of the form $3k+1$ is itself of the form $3k+1$. But $N$ is not of the form $3k+1$. So some prime $p$ of the form $3k+2$ divides $N$. We already saw that $p$ cannot be one of $q_,\dots,q_n$. It follows that given any collection $\{q_1,\dots,q_n\}$ of primes of the form $3k+2$, there is a prime $p$ of the same form which is not in the collection. Thus the number of primes of the form $3k+2$ cannot be finite.