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

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.