Hi, I am having trouble understanding this proof. Statement If pn is the nth prime number, then pn [tex]\leq[/tex] 22n-1 Proof: Let us proceed by induction on n, the asserted inequality being clearly true when n=1. As the hypothesis of the induction, we assume n>1 and the result holds for all integers up to n. Then pn+1 [tex]\leq[/tex] p1p2...pn + 1 pn+1 [tex]\leq[/tex] 2.22...22n-1 + 1 = 21 + 2 + 22+ ...+ 2n-1 Recalling the indentity 1 + 2 +22+ ...+2n-1=2n-1 Hence pn+1 [tex]\leq[/tex] 22n-1+1 But 1 [tex]\leq[/tex] 22n-1 for all n; whence pn+1 [tex]\leq[/tex]22n-1+22n-1 =2.22n-1 =22n completing the induction step, and the argument. What I don't understand is why the proof uses p1, p2, etc as powers of two. What is the nature of the pn? Are they prime or what? Why use powers?