Primes in Beatty sequence

For a polynomial g(x) of deg k >= 2 with integer coefficients and positive integer leading coefficient, we prove an upper bound for the least prime p such that g( p) is in non-homogeneous Beatty sequence {left perpendicular alpha n + beta right perpendicular: n = 1, 2, 3,...}, where alpha, beta is an element of R with alpha > 1 is irrational and we prove an asymptotic formula for the number of primes p such that g(p) = left perpendicular alpha n + beta right perpendicular. Next, we obtain an asymptotic formula for the number of primes p of the form p = left perpendicular alpha n + beta right perpendicular which also satisfies p equivalent to f (mod d), where f, d are integers with 1 <= f < d and ( f, d) = 1.