Dear number theorists,


    If p, p + 2 and pi(p) (the number of primes not exceeding p) are all 
prime, then I'd like to call {p, p + 2} a super twin prime pair. Here I 
report my following discovery.


    Super Twin Prime Conjecture (Feb. 5, 2014). Each n = 3, 4, ...  can 
be written as k + m with k and m positive integers such that

      p(k) + 2    and   p(p(m)) + 2

are both prime, where p(j) denotes the j-th prime.


    For example, 22 = 20 + 2,  {p(20), p(20) + 2} = {71, 73} is a twin 
prime pair and

      {p(p(2)), p(p(2)) + 2} = {p(3), p(3) + 2} = {5, 7}

is a super twin prime pair.

    The conjecture implies that there are infinitely many super twin 
prime pairs. In fact, if all those positive integers m with p(p(m)) + 2 
prime are smaller than an integer N > 2, then by the conjecture, for 
each j = 1, 2, 3, ... there is a positive integer k(j) in the interval 
((j-1)*N, j*N) with p(k(j)) + 2 prime, and hence

     sum_{j>0} 1/p(k(j)) >= sum_{j>0} 1/p(j*N)

which contradicts Brun's theorem on twin primes.

    I have verified the conjecture for n up to 2*10^8. For related data 
and graphs concerning the representation function for this conjecture, 
one may visit

       http://oeis.org/A218829   and    http://oeis.org/A218829/graph .


    I consider the above conjecture quite challenging and mysterious. In 
my opinion, it might never be solved by human beings.


    By the way, I also conjectured that any integer n > 7 can be written 
as k + m with k and m positive integers such that

       phi(k) - 1, phi(k) + 1  and  p(p(p(m))) - 2

are all prime, where phi(.) denotes Euler's totient function. See
http://oeis.org/A237253 .


    Any comments are welcome!


      Zhi-Wei Sun (Nanjing Univ., China)
      http://math.nju.edu.cn/~zwsun