TITLE: What is wrong with this proposed proof of the twin prime conjecture?
QUESTION [11 upvotes]: I was thinking on the twin prime conjecture, that there are an infinite number of twin primes...   I came up with a proof.  I have to think that it is incomplete or wrong, because many great minds have thought on this previously. I can't see the issue, so I thought I would raise it in  a broader forum.  What is wrong with this proof? 
1) A number n is prime iff n mod p is non zero for every prime number  1 < p < n
This is easy to prove from the definition of a prime number and mod. It just says that n is not equally divisible by another prime, making it prime.
Let $p_n$ = the nth prime $p_0 = 1$, $p_1 = 2$ ...  
Consider $N_n = \Pi_0^n p_n$.  
It is easy to see that  $N_n  \mod  p_j = 0 $  for all primes  $0 < j < n$
$(N_n + 1)  \mod  p_j = 1 $  for all primes  $0 < j < n$, and therefore, from #1 must be prime
$(N_n - 1)  \mod p_j  = (p_j - 1) $ for primes  $0 < j < n$, and therefore, from #1 must be prime as well
The set $p_n$ has infinite members (as shown by Euclid) ,  so there are infinite $N_n$. Therefore there are an infinite set of twin prime numbers $(N_n-1,N_n+1)$ .

REPLY [4 votes]: $N_7+1 = 510511 = 19 ⋅ 97 ⋅ 277$, $N_7-1 = 510509 = 61 ⋅ 8369$ is the first example where both numbers are not primes. I would suggest that not only is $(N_k-1, N_k+1)$ not always a twin prime pair, but that this would actually be quite rare. 
According to http://primes.utm.edu/top20/page.php?id=5 $N_k+1$, $N_k-1$ have been tested for k ≤ 100,000 with very few primes found, and with no twin primes found beyond the pair (2309, 2311).