On the A $\mathcal {A}$ script upper A-transcendence of a Champernowne-type constant

Abstract Let p n $p_n$ p Subscript n denote the n th prime. We prove that, for every nonzero polynomial f ( x ) ∈ Z [ x ] $f(x)\\in \\mathbb {Z}[x]$ f left parenthesis x right parenthesis element of double struck upper Z left bracket x right bracket , there exist infinitely many positive integers n such that p n ∤ f ( n ) $p_n\\nmid f(n)$ p Subscript n Baseline does not divide f left parenthesis n right parenthesis .