Sieves for Generating Prime k-tuples and Counting Functions

Sieve methods for generating prime k-tuples are disclosed. By direct counting based on indication of the Chinese remainder theorem, the actual counts of different prime k-tuples are calculated after the sieve procedures. The results, with mathematical approximations for analytic expressions but no conjectures, seem to match well with those of the prior conjectures which were mostly based on probabilities and complex methods. It’s found that there must be none, a single one, or infinitely many prime k-tuples of any particular form. No other finite counts are possible. It happens that the count of any prime pair, with the difference between the two elements equal to an even number, is infinite.