On a Stricter Twin Primes Conjecture, and on the Polignac's Conjecture in general

Abstract: The Polignac's Conjecture, first formulated by Alphonse de Polignac in 1849, asserts that, for any even number M, there exist infinitely many couples of prime numbers P, P+M. When M = 2, this reduces to the Twin Primes Conjecture. Despite numerical evidence, and many theoretical progresses, the conjecture has resisted a formal proof since. In the first part of this paper, we investigate a stricter version of the conjecture, expressed as follows: ''Let $p_{n}$ be the n-th prime. Then, there always exist twin primes between $(p_{n}-2)^{2}$ and $p_{n}^{2}$ ''. To justify this conjecture, we formulate a prediction (based on a double-sieve method) for the number of twin prime pairs in this range, and compare the prediction with the real results for values of $p_{n}$ up to 6500000. We also analyse what should happen for higher values of $p_{n}$. In the second part, we investigate the validity of the general Polignac's Conjecture. We predict the ratio of the number of solutions for any value of M divided by the number of solutions for M = 2, and explain how this ratio depends on the factorization of M. We compare the predictions with the real values for M up to 3000 (and for the special case 30030) in the range of from the 1000000-th prime to the 21000000-th prime.