Infinite sets of primes, admitting Diophantine representations in eight variables

The existence of infinite sets of primes which can be repesented as the sets of positive values of some polynomials in a small number of variables is discussed. (All variables range over positive integers.) It is proved (noneffectively) that there exists such a set, which has a representation with eight variables. This number of variables is smaller than in the best universal construction known today, which is ten. Also, some improvements of well-known technical lemmas are given. Bibliography: 16 titles.