Sieves for Generating Prime k-tuples and Counting Functions

Sieve methods for generating prime k-tuples are disclosed. Based on indication of the Chinese remainder theorem, every natural number is expressed by a unique residue vector or tuple with respect to all prime numbers. Sieves are then implemented to remove the undesired residue classes, resulting in infinitely many prime k-tuples of each admissible form, such as twine primes, cousin primes, sexy primes, etc., except the 1-tuple prime numbers. It is also proved that there are infinitely many Chernick’s Carmichael numbers. By direct counting, 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.