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

Allprimescanbeindexedbyk,asprimesmustbeintheform 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.