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Statement

Here HLC denotes the inequality $\pi(m+n)\le\pi(m)+\pi(n)$ for all integers $m,n\ge2$ (display (1.2), p. 1), and $p_r$ is the $r$-th prime.

Proposition 2.4 (p. 2, quoted). "Let $N_0=1.7\times10^9$. Then the HLC holds for all integers $m,n\ge2$ satisfying $m+n\le39\,708\,229\,123=p_{N_0}$."

This is a finite verification.

Proof pointer

P. 2. Segal's criterion (Lemma 2.1, p. 2) says that HLC holds if and only if $p_k\ge p_{k-q}+p_{q+1}-1$ for all integers $k\ge3$ and $1\le q\le(k-1)/2$, display (2.1). By Segal's Lemma 2.2, if HLC fails, the least failing value of $m+n$ is the least $p_k$ at which (2.1) fails, and Panaitopol's Lemma 2.3 restricts the check to $k\ge9680$ and $34\le q\le(k-1)/27$. With these lemmas and a computer, Panaitopol had reached $m+n\le p_{250000}=3\,497\,861$; the paper says only that the proposition extends this computation. Its acknowledgement (p. 9) credits Thomas Leßmann with a C++ program written to verify Proposition 2.4. No further account of the computation is given.

Read depth

Claims checked: the statement and Lemmas 2.1 to 2.3 were read clause by clause on the pages of the copy named on the source card. The computation was not repeated, and the paper's account of it is only the attribution above. Nothing here is independently reviewed.

Dependencies

• Lemmas 2.1 and 2.2, cited from S. Segal, Trans. Amer. Math. Soc. 104 (1962), 523--527, Theorems I and II (see its card).
• Lemma 2.3, cited from L. Panaitopol, Rev. Roumaine Math. Pures Appl. 46 (2001), 465--470.

Source. Christian Axler, "Some Results on a Conjecture of Hardy and Littlewood," arXiv:1909.12625v2 (2019), the edition read for the source card.

Bears on

• Problem 855: the proposition proves the problem's inequality for every pair of integers at least $2$ with sum at most $39\,708\,229\,123$. A finite range cannot decide a statement about all large arguments.