Call a positive integer good if either N=1 or N can be written as product of even number of prime numbers, not necessarily distinct.
Let P(x)=(x−a)(x−b), where a,b are positive integers.(a) Show that there exist distinct positive integers a,b such that P(1),P(2),⋯,P(2010) are all good numbers.
(b) Suppose a,b are such that P(n) is a good number for all positive integers n. Prove that a=b. number theoryprime numbersnumber theory unsolved