On Equation Of Ab = (nα)

Number theory is main about integers. The distribution of prime numbers has been the key of number theory for many years. Recently many papers have been written on prime number theory, which has correspondingly improved the system of number theory.One of the most odd problem in prime number theory is to represent infinitely many prime numbers by polynomials. Though this problem has been completely solved by linear polynomials by the Dirichlet theorem on primes in arithmetic pro-gressions, until now it remains open for polynomials of degree greater than 1 and it seems to be extremely difficult. Iwaniec[1] has proved that there exists an infinity of integers n such that n2+1 has at most two prime factors.As another direction of approaching this problem, Piatetski-Shapiro[2] has pro-posed to investigate the prime numbers of the form [nc], where 1>N1+2ε, (4) then the equation(3)is solvable.Let S be the number of solutions of(3),then we haveFrom above results,we consider a variant of the Piatetski-Shapiro prime num-ber problem,with the following result:Theorem2 Suppose k>11 is fixed,c0=1/25(28-31/(k-1-log4/log3)-100η),for 110 is fixed,c1=1/565(636-636/(k+1-log4/log3)).There exsit N0=N0(c)>0 andδ=δ(c)>0,such that for N>N0 and some sets A,B(?)(Nc/2, (2N)c/2], for 1