Number Theory : Primes not in Twin Primes I was working through some basic number theory questions , when I came across :


  Show that there are infinitely many primes that are not one of the primes in a pair of twin primes


How can I go about solving it ? I have absolutely no idea ...
All primes $p$ and $q$ of the form $p = 23+30n$, $q = 37 + 30n$ where $n$ is natural, are unpaired.  

To proof this, think of all of the factors of 30 and why 25 and 35 will not be prime. Know also, all prime pairs greater than 3, like 5,7 use the forms 6n-1, 6n+1 to make a pair.
With the exception of 7, numbers ≡ {7, 23} mod 30, which sequence as 7 {+16+14} {repeat ... ∞} cannot possibly be twin primes given their closest possible proximity to other prime numbers is +4 and -4, respectively. A modulo 30 factorization wheel, which is populated by n ≡ {1, 7, 11, 13, 17, 19, 23, 29} (mod 30) (aka, natural numbers not divisible by 2, 3, or 5), parses all twin prime candidates > (5,7) into 3 distribution channels, and makes this clear:


n ≡ {11,13} mod 30 (at 132° &amp; 156°)
n ≡ {17,19} mod 30 (at 204° &amp; 228°)
n ≡ {29, 1} mod 30 (at 348° &amp; 12°).


Ref: primesdemystified.com/twinprimes