---

Statement

The Dickman function $\varrho$ (p. 42) is the continuous function with $\varrho(u)=1$ for $0\le u\le1$ and $\varrho(u)=\varrho(N)-\int_N^u v^{-1}\varrho(v-1)\,dv$ for $N<u\le N+1$, $N=1,2,\ldots$.

Theorem 2 (p. 43). Let $\varepsilon$ and $\theta$ be real numbers with $0<\varepsilon<1$ and $0<\theta<1$. Let $k$ and $l$ be positive integers with $2\le l\le\theta\log k$, where $k$ exceeds a number effectively computable in terms of $\varepsilon$ and $\theta$. Then there are distinct positive integers $a_1,\ldots,a_k$ and distinct non-negative integers $b_1,\ldots,b_l$ with

$$\text{(7)}\qquad P\Bigl(\prod_{i=1}^k\prod_{j=1}^l(a_i+b_j)\Bigr)<k^{h(\theta)+\varepsilon}, \qquad h(\theta)=\min_{u\ge1}\Bigl(\frac{1-\theta\log\varrho(u)}{u}\Bigr).$$

Here $P(n)$ is the greatest prime factor of $n$.

Remarks of the paper (pp. 43--44). The minimum defining $h(\theta)$ is attained, and $h(\theta)<1$ for $0<\theta<1$, so (7) improves by a power on the trivial bound $k+l$ given by $a_i=i$, $b_j=j-1$. For $\theta\le1/6$, Buchstab's lower bound for $\varrho$ gives $h(\theta)\le\theta\bigl(1+\log(1/\theta)+\log\log(1/\theta)+6\log\log(1/\theta)/\log(1/\theta)\bigr)$. The bound (7) also holds with $\omega$, the number of distinct prime factors, in place of $P$. In the introduction (p. 39) the authors state that it follows from Theorem 2 that, even for $l$ of the form $\delta\log k$ with $0<\delta<1$, the bound (2), $P(a+b)>C_3\log k\log\log k$ (p. 38), cannot be replaced by $k^{1-\varepsilon}$ for every $\varepsilon>0$ and $k\ge k_0(\delta,\varepsilon)$.

Conjectures stated (pp. 39 and 44). For $l>\log k$ and every $\varepsilon>0$ the authors conjecture that some $a\in A$, $b\in B$ have $P(a+b)>k^{1-\varepsilon}$ for $k\ge k_1(\varepsilon)$ (p. 39); and that there is no positive real $\gamma<1$ with arbitrarily large $l,k$, $l>\log k$, admitting distinct positive $a_1,\ldots,a_k$ and distinct non-negative $b_1,\ldots,b_l$ with $\omega\bigl(\prod_{i,j}(a_i+b_j)\bigr)<(\pi(k+l))^\gamma$ (p. 44).

Proof pointer

Pp. 47--48. Lemma 6 (p. 46) is the analogue of Lemma 3 with the smooth-number count taken from Dickman's asymptotic $\psi(x,x^{1/u})\sim x\varrho(u)$ (Lemma 5, p. 46): applying Lemma 1 to the $N^{1/u}$-smooth integers up to $N$ gives about $(N/l)\varrho(u)^l$ integers $a$ with every $a+b_j$ smooth. The proof of Theorem 2 takes $u=u_0$, a point where $(1-\theta\log\varrho(u))/u$ is minimal, and $N=\lceil kl((1-\varepsilon/2)\varrho(u_0))^{-l}\rceil$.

Read depth

Claims checked: the definition of $\varrho$, the statement and the remarks above were read clause by clause on the page images of the print. The proof was followed for the outline above and is not independently verified.

Dependencies

• Lemma 1 (p. 39), through Lemma 6 (p. 46).

Source. P. Erdős, C. L. Stewart and R. Tijdeman, Some diophantine equations with many solutions, Compositio Mathematica 66 (1988), 37--56; the edition read is named on the source card.

Bears on

No problem page of this corpus.