A Symmetric Uniform Formula and Sole Index Method for Sieving (Twin) Primes

All primes can be indexed by k, as primes must be in the form of 6k+1 or 6k-1. In this paper, we explore for the set of k such that either 6k+1 or 6k-1 is not a prime. Our method provides a uniform formula for k that can sieve primes and twin primes as well. The uniform presents symmetry in terms of k that works as single index in sieving. We also propose a new conjecture that is equivalent to Twin Prime Conjecture but possibly be easier to approach by merely exploring of sole index in term of k. The uniform formula for prime sieving are as follows: k ∈ S_l ⇒ 6k-1 ∉ P, where s_l = [-I]6I+1 = [I]_6I-1 \min([I]_6I-1), I ∈ N. ∈ s_r ⇒ 6k+1 ∉ P, where s_r = [-I]_6I-1 ∪ ([I]_6I+1 \min([I]_6I+1)), I ∈ N.