Note on the Prime Number Theorem
نویسندگان
چکیده
Proof. First of all, we prove that if pn is the nth prime number then we have that pn ≤ 2 n−1 . Since there must be some pn+1 dividing the number p1p2 · · · pn− 1 and not exceeding it, it follows from the induction step that pn+1 ≤ 2 0 2 1 · · · 22n−1 = 220+21+···+2n−1 ≤ 22n . If x ≥ 2 is some real number, then we select the largest natural number n satisfying 22n−1 ≤ x, so that we have that 2 n > x. Hence we conclude that π(x) ≥ n ≥ 1 ln 2 · ln (
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