On Some Infinite Series Related to the Twin Primes

It is shown that all positive integers can be divided into numbers that can lead to a pair of twin primes by a simple algebraic operation and numbers that cannot. The paper devises a formula for finding the numbers in each category, and shows that the numbers in the second category can be arranged in infinite groups and super-groups with an inner symmetry, a precise interval length and a well-defined number of terms. After presenting some of their properties, the paper analyzes a relatively small interval remained after subtracting certain numbers that belong to the second category from the set of positive integers in the interval. Because all terms in the interval depend on a single prime P to fall into one category or another, the fraction of terms from the second category is about 2 / P of the total number of terms. The rest must belong to the first category.