Analyzing twin primes, Goldbach's strong conjecture and Polignac's conjecture

Abstract

Here we analyze three well-known conjectures: (i) the existence of infinitely many twin primes, (ii) Goldbach's strong conjecture, and (iii) Polignac's conjecture. We show that the three conjectures are related to each other. In particular, we see that in analysing the validity of Goldbach's strong conjecture, one must consider also the existence of an infinite number of twin primes. As a consequence of how we approach this analysis, we also observe that if this conjecture is true, then so is Polignac's conjecture. Our first step is an analysis of the existence of infinitely many twin prime numbers. For this, using the formula 4((n−1)!+1) ≡−n (mod n(n + 2)) – satisfied if and only if (n, n + 2) are twin primes –, together with Wilson's theorem, we obtain conditions that must be met for two numbers to be twin primes. Our results, obtained from an analytic and functional study, lead us to conclude that there may exist infinitely many twin primes. Next, we consider the validity of Goldbach's strong conjecture. After showing that the conjecture is true for the first even numbers, we notice a pattern that we analyze for any even number, reducing it to three cases: (i) when the even number 2n is two times a prime number n; (ii) when the even number 2n is such that n=2m, with 2m-1 and 2m+1 twin primes; (iii) all other cases, i.e., for any 2n even number ∀n∈N with n > 1, n prime or not, with independence of n=2m being 2m-1 and 2m+1 twin primes or not. In this last case, we show that one can always find a certain r∈N such that 1 < r < n satisfying that n − r and n + r are primes, so that their sum is 2n. In this case, we use the reduction to absurd method, and our results lead us to conclude that Goldbach's strong conjecture is true to the best of our calculations, and Polignac's conjecture as well.