In this paper two generator algorithms for primes and twin primes are proposed, both of them are based on the the divisibility properties of binomial expressions. The mathematical foundation lies in the connection that exists between binomial expressions and the number of carries that result in the sum in different bases of the variables that form the binomial expression. The first one besides decomposing an integer into its prime factorization, generates the primes, while the second one besides answering if given two twin integers, whether they are twin primes or not by just verifying one condition, generates the twin primes. The proposed procedures assume the knowledge of the primes up to some fix number. The mathematical approach applied is novice, and both algorithms are new.