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Statement

Notation as on the Theorem I page: $P_i$ is the $i$-th prime, (1) is $\pi(x+y)\leq\pi(x)+\pi(y)$, and (2) is $P_n\geq P_{n-q}+P_{q+1}-1$ for $n\geq3$ and integers $1\leq q\leq(n-1)/2$.

Theorem II (p. 523; quoted). "If (1) is false for some integer $x+y$, then the smallest such value of $x+y$ is the smallest value of $P_n$ for which (2) is false." The restatement on p. 526 reads "the smallest $P_n$" in place of "the smallest value of $P_n$".

Here $x,y\geq2$, as throughout the paper, and "(2) is false" for $P_n$ means that (2) fails for some integer $q$ with $1\leq q\leq(n-1)/2$.

Lemma V (p. 526). If (1) fails for some $x,y\geq2$, the least value of $x+y$ at which it fails is prime.

The computation (p. 527). Inequality (2) was checked by machine (an IBM 1620 at Wesleyan University, programmed by William Jeffreys from D. N. Lehmer's tables of primes) and found to hold for $n\leq9679$, that is for $P_n\leq101{,}081$. With Theorem II the paper concludes that (1) holds whenever either variable is at most $132$ (Schinzel and Sierpiński, cited) or $x+y\leq101{,}081$. Here $P_{9679}=101{,}081$.

Source. Sanford L. Segal, On $\pi(x+y)\leq\pi(x)+\pi(y)$, Trans. Amer. Math. Soc. 104 (1962), no. 3, 523--527, doi:10.1090/s0002-9947-1962-0139586-4: Theorem II stated on p. 523, restated on p. 526 and proved on pp. 526--527; Lemma V and its proof on p. 526; the computation on p. 527. The edition read is identified on the source card.

Read depth. Claims checked: the statements of Theorem II and Lemma V and the report of the computation were read clause by clause on the printed pages; the proofs were read. The computation was not repeated. Nothing here is independently reviewed.

Proof pointer

Lemma V (p. 526): if the least failing sum $Z_0=X_0+Y_0$ were composite, comparing $X_0+Y_0$ with the largest prime below it and $Y_0$ with the largest prime not exceeding it produces, in each of two cases, a failing pair with a smaller sum.

Theorem II (pp. 526--527): by Lemma V the least failing sum is a prime $P_n=X_0+Y_0$ with $Y_0>X_0$, and $n\geq3$. Choosing $q$ so that $P_{n-q-1}$ is the largest prime not exceeding $Y_0$, the failure at the pair $X_0,Y_0$ gives $X_0\leq P_{q+1}-1$, the paper's (14), hence $P_n\leq P_{q+1}+P_{n-q}-2$, its (15), so (2) fails at $P_n$; the paper then argues that $q$ may be taken at most $(n-1)/2$. The printed proof does not spell out the reverse comparison, that no smaller prime fails (2). It follows from the analysis in the proof of Theorem I (this paragraph is the corpus's reasoning, not the paper's): if (2) fails at $P_m$ for an admissible $q$, then $q\geq2$ (for $q=1$, (2) reads $P_m\geq P_{m-1}+2$), so the even values $P_{m-q}+P_q$ and $P_{m-q}+P_{q+1}-2$ are excluded and $P_m$ satisfies (9) or $P_m\leq P_{m-q}+P_q-1$, and either gives a failing pair with sum $P_m$, namely the pair of Lemma IV in the first case and $x=P_m-P_{m-q}$, $y=P_{m-q}$ in the second.

Dependencies

• Lemma IV (p. 525), through the reverse comparison above.
• Lemma V (p. 526), stated above.
• A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), 201--206 (the paper's reference 3), for the range where one variable is at most $132$.

Bears on

• Problem 855: Theorem II and the computation exclude every violation with $x+y\leq101{,}081$. A finite range does not decide the problem's question for large $x$ and $y$, and the paper claims nothing beyond it.