---

Statement

Theorem II (p. 609). For $n$, with no range stated in the paper,

$$\pi(n)>\log_2\Bigl(\frac n3\Bigr),$$

where, as the paper states, $\pi(n)$ denotes the number of primes $<n$. The paper presents the theorem as a corollary of Theorem I.

The deduction in Section 4 (p. 610) says that the prime divisors of the sums it uses are the primes $\le n$, and so applies Theorem I with the primes up to and including $n$; for prime $n$ that count exceeds the strict count of the statement by one. This page records the discrepancy between the statement's $<n$ and the deduction's $\le n$ and does not resolve it.

Source. Paul Erdős and Paul Turán, On a problem in the elementary theory of numbers, Amer. Math. Monthly 41 (1934), 608-611: Theorem II on p. 609, its deduction in Section 4 on p. 610. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the deduction were read clause by clause on the printed pages. Nothing here is independently reviewed.

Proof pointer

Section 4, p. 610. Take $a_v=v$ for $v=1,\ldots,\lceil n/2\rceil$. Every two-term sum is at most $n$, so its prime factors are among the primes up to $n$; Theorem I then forces $n/2<3\cdot2^{\pi(n)-1}$, which rearranges to the stated inequality.

Dependencies

Theorem I of the same paper.

Bears on

None of the corpus's problem pages.