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Statement

Theorem 5 (p. 53). Let $\varepsilon>0$, let $2=p_1,p_2,\ldots$ be the primes in order, and let $l\ge2$ be an integer. There is a number $s_0(\varepsilon,l)$, effectively computable in terms of $\varepsilon$ and $l$, such that for every integer $s\ge s_0(\varepsilon,l)$ there is a monic polynomial $F(X)$ of degree $l$ with distinct roots and rational integer coefficients for which the equation

$$\text{(15)}\qquad F(x)=p_1^{z_1}\cdots p_s^{z_s}$$

has at least

$$\text{(16)}\qquad \exp\Bigl\{(l^2-\varepsilon)\frac{s^{1/l}}{(\log s)^{(l-1)/l}}\Bigr\}$$

solutions in non-negative integers $x,z_1,\ldots,z_s$.

Remark (p. 54). The polynomial built has only rational integer roots; the authors state that a comparable lower bound remains open when, for instance, $F$ is irreducible over the rationals.

Context in the paper. For an irreducible binary form $F\in\mathbb Z[X,Y]$ of degree $n\ge3$ with non-zero discriminant, the introduction (p. 37) states Evertse's bound $\exp(n^3(4s+7))$ for the number of coprime pairs $x,y$ with $F(x,y)$ composed of primes from $S$, and the authors state (p. 38) that it follows from Theorem 5 that this bound cannot be replaced by $\exp(n^2s^{1/n}/\log s)$, not even when $F$ is a polynomial in one variable.

Proof pointer

Pp. 53--54. Apply Lemma 3 (p. 40) with $c=1$, $f(x)=(\log x)/l$ and $N=\lfloor\exp\{(l-\delta)(s\log s)^{1/l}\}\rfloor$. It gives $a_1,\ldots,a_m$ and $b_1,\ldots,b_l$ with every $a_i+b_j$ free of primes above $(1-\delta/l)^ls\log s$, which is at most $p_s$ by the prime number theorem; then $F(X)=(X+b_1)\cdots(X+b_l)$ and $x=a_i$ give the solutions.

Read depth

Claims checked: the statement and the remark 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 3 (p. 40).

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

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