SOME DIOPHANTINE EQUATIONS AND INEQUALITIES WITH PRIMES

摘要

We consider the solutions to the inequality vertical bar p(1)(c) +center dot center dot center dot+ P-S(C) -R vertical bar 1, c (sic) N and eta is a small positive number; R is large). We obtain new ranges of c for which this has many solutions in primes p1, . . . ,p(s) for s = 2 (and 'almost all' R), s = 3, 4 and 5. We also consider the solutions to the equation in integer parts [p(1)(c)] +center dot center dot center dot+ P-S(C) = r where r is large. Again c > 1, c (sic) N. We obtain new ranges of c for which this has many solutions in primes, for s = 3 and 5.