Volume 14, 2008, Number 3

Primes were separated according to the right-end-digits (REDs) and classes in the modular ring Z₆. The primes are given by jR, where j is the number of consecutive integers with RED = R (for p = 37, R = 7 and j = 3, and so on). The rows of j in classes ̅2₆, ̅4₆ ⊂ Z₆, that contain primes, are found to have the form ½n(an ± 1), a = 1, 3, 5. A total of 499 primes were generated for n = 1 to 80 for RED = 7. Similar results apply to REDs 1, 3 and 9. A scalar characteristic was detected in the row structure of j.

In the paper, the polynomials with integer coefficients are considered and a hypothetical sufficient condition for these polynomials to take infinitely many primes as their values is proposed. That provides an alternative equivalent variant of the famous Dirichlet’s theorem for infinitely many primes in arithmetic progressions. Also an interesting analogy between the behaviour of polynomial’s zeros and the integers for which the polynomial takes prime values is noted.

This paper considers some properties of rising and falling factorials by analogy with some classic results in number theory for cycles of binomial coefficients.