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Statement

Theorem 4 (p. 49). Let $\varepsilon>0$. There is a number $s_0(\varepsilon)$, effectively computable in terms of $\varepsilon$, such that for every integer $s>s_0(\varepsilon)$ there is a set $S$ of primes with $|S|=s$ for which the equation $x+y=z$ has at least $\exp\bigl((4-\varepsilon)(s/\log s)^{1/2}\bigr)$ solutions in coprime positive integers composed of primes from $S$.

Context in the paper (pp. 48--49). The authors compare this with Evertse's upper bound of $3\times7^{2s+3}$ solutions in coprime integers for $ax+by=cz$, and call that bound not far from best possible.

Conjecture (p. 49). On the basis of a heuristic computation the authors conjecture that for every $\varepsilon>0$, when $S$ is the set of the first $s$ primes, $x+y=z$ has at least $\exp(s^{(2/3)-\varepsilon})$ solutions in coprime positive integers composed of primes from $S$ for $s>C_1(\varepsilon)$, and that for any set $S$ of $s$ primes the number of solutions is at most $\exp(s^{(2/3)+\varepsilon})$ for $s>C_2(\varepsilon)$.

Corollary (p. 52). The authors note as an immediate consequence that for $\varepsilon>0$ and $s>s_0(\varepsilon)$ some set $S=\{p_1,\ldots,p_s\}$ of primes makes $xy(x+y)=p_1^{z_1}\cdots p_s^{z_s}$ have at least $\exp\bigl((4-\varepsilon)(s/\log s)^{1/2}\bigr)$ solutions in non-negative integers $x,y,z_1,\ldots,z_s$ with $\gcd(x,y)=1$.

Proof pointer

P. 51. Apply the first part of Theorem 3 with $s_1$ in place of $s$ and a small $\delta$ in place of $\varepsilon$, where $s_1=[s(1+\delta)^{-1}]$. The resulting difference $k_1$ has at most $4(s_1/\log s_1)^{1/2}$ prime factors, so the primes up to $p_{s_1}$ together with those dividing $k_1$ form a set $S$ with $|S|\le s$. Each solution of $x-y=k_1$ gives a solution of $k_1+y=x$ in integers composed of primes from $S$, and dividing out the common factor gives distinct coprime solutions of $x+y=z$.

Read depth

Claims checked: the statement, the conjecture and the corollary 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

• Theorem 3 (p. 49).

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.