A method for operating a computer for generating a comprehensive and exclusive list of prime numbers which are less than or equal to a selected large integer M, the method including steps of operating the computer to establish a row head series R, to establish an integer value A, to establish an integer value L, to establish a series of integer rows based on R and A, to cyclicly generate and regenerate the series of integer rows based upon cyclicly redefined values of R and A, and to perform prime and non-prime labeling of members of a very large finally generated series of integer rows in the manner of the method of the Sieve of Eratosthenes.