Pmath 440/640 Analytic Number Theory

Yes, but such a formula is complicated. For example, is there a polynomial f ∈ Z[x] for which f(n) = pn? f(x) = anx n + · · ·+ a1x+ a0 f(a0) = ana0 n + · · ·+ a1a0 + a0 so a0 | f . Suppose q is prime and f(n) = q. Then q | f(n + kq) for each k ∈ Z. So, in particular, we see that if f(m) is prime for each positive integer m, then f is a constant. In particular, f(x) = q for some prime q. The polynomial n + n + 41 is prime for n = 0, 1, . . . , 39. Further, (n− 40) + (n− 40) + 41 is prime for 0 ≤ n ≤ 79.