Infiniteness of non-twin primes.

Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are infinitely many primes $p$ such that $p + 2$ is NOT prime.

Solution 1:

Let $p\gt 3$ be prime. If $p+2$ is not prime, we are happy. If $p+2$ is prime, then $(p+2)+2$ is not, since one of $x,x+2,x+4$ is divisible by $3$.

Added: Dolda2000 noted that a more interesting question is whether there are infinitely many primes that are not members of a twin pair. For this we can use the fact that there are infinitely many primes of the form $15k\pm 7$. If $p$ is such a prime, then one of $p-2$ or $p+2$ is divisible by $3$, and the other is divisible by $5$, so if $p\gt 7$ then neither $p-2$ nor $p+2$ is prime.

Solution 2:

Suppose the contrary. This would mean that from some point on, all odd numbers are prime numbers.

Take two sufficiently large odd numbers. Multiply them together. The result is a composite number which is odd, which contradicts the hypothesis that all sufficiently large odd numbers are primes.

Solution 3:

Dirichlet's Theorem guarantees the existence of infinitely many primes of the form $p = 3n+1$, and for each of these, $p+2$ is a multiple of 3.

Solution 4:

For any integer $x$, either $x$ or $x+2$ or $x+4$ is divisible by $3$, so if $p$ is a prime $> 3$, either $p$ or $p+2$ is an example.

Solution 5:

Euler proved that the sum of the reciprocals of the primes diverges. Brun proved that the sum of the reciprocals of the twin primes converges. The sum of the reciprocals of the non-twin primes must diverge, so there are infinitely many.