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Claim. P. Dusart, Sur la conjecture $\pi(x+y)\le\pi(x)+\pi(y)$, Acta Arith. 102 (2002), no. 4, 295--308, Théorème 1 (p. 296): for all real $x,y$ with

$$2\le x\le y\le\frac75x\log x\log\log x,$$

$\pi(x+y)\le\pi(x)+\pi(y)$. The proof splits at the ratio $y/x=109$. Proposition 3 (p. 304) proves the inequality for real $x,y\ge3$ with $1/109\le y/x\le109$, through the prime-index inequality $p_k+p_l\le p_{k+l-1}$ of Proposition 2 (p. 301), whose small indices are checked by computer (Lemma 4, pp. 302--304). Proposition 1 (p. 299) covers $109\le y/x\le\frac75\log x\log\log x$ for $x\ge0.4\cdot10^{11}$, from explicit two-term bounds on $\pi$. Proposition 4 (p. 305) deduces that the pairs $x,y\le X$ failing the inequality make up at most a proportion $5/(7\log X\log\log X)$ of the square. The paper is compiled at its card, which holds the published article.

Covers. Every pair whose smaller entry $u\ge2$ and larger entry $v$ satisfy $v\le\frac75u\log u\log\log u$. Not covered: the pairs with $v>\frac75u\log u\log\log u$, which include pairs with both entries arbitrarily large, so the problem as posed is not settled.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Acta Arithmetica 102 (2002), a refereed journal (received 7 May 1999, revised 18 April 2001, as the article prints). The journal volume gives the year 2002 and no month or day, so the page's date carries the first day of that year.