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Statement

Lemma 2.17 (pp. 640--641). "Given constants $b$, $c>0$, then for almost all integers $x$

$$d(x+y)<b^{-1}(2c)^{-y}(\log x)^{3y/4};\quad y=3,4,\cdots\text{"}$$

The display is the paper's (2.18), printed at the top of p. 641; $d$ is the number-of-divisors function. One $x$ must satisfy the bound for all $y\ge3$ at once. "Almost all" is used in the sense the proof makes precise: the set of exceptional $x$ below $N$ is $o(N)$ (p. 641).

Source. P. Erdős, E. G. Straus, Some number theoretic results, Pacific J. Math. 36 (1971), no. 3, 635--646; Lemma 2.17 on pp. 640--641, its proof on p. 641. The copy read is identified on the source card.

Read depth. Claims checked: the statement and the exponent $3y/4$ were read on the page images of pp. 640--641 at high resolution; the proof was read for structure, not checked. Nothing here is independently reviewed.

Proof pointer

Page 641. For large $x$ the bound is automatic once $y>2\log x$, since the right side then exceeds $x+y\ge d(x+y)$; so a failure needs some $3\le y\le2\log x$. If a positive proportion of $x\le N$ failed, some single $y$ would carry a share $1/(2\log N)$ of them, and the large divisor values at the shifted points would push $\sum_{n\le N}d(n)$ to at least a constant times $N(\log N)^{5/4}$ (the case $y=3$ printed on p. 641), contradicting the Dirichlet divisor theorem.

Dependencies

The Dirichlet divisor theorem $\sum_{n\le N}d(n)\sim N\log N$, the paper's (2.15) (p. 640).

Bears on

• #258: an ingredient of Lemma 2.14, and through it of Theorem 2.23; on its own it proves no case of the problem.