Show that for every positive integer N there is an even number K so that there are more than N pairs of successive primes such that K is the difference between these successive primes (Hint: Use prime number theorem)
So, this is how far I've gotten..
Consider a list of primes and consider the difference of the successive primes
p_t - p_(t-1) = K
p_(t-1) - p_(t-2) = K
p_2 - p_1 = K
From here, I've been told to consider the sum of these differences and I should arrive to a conclusion that should contradict the prime number theorem.