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Jun 12, 2020 at 2:32 comment added Joe Silverman @Lucia Thanks. That's good to know. Here's the JNT referece: Konyagin, S.; Soundararajan, K. Two S-unit equations with many solutions. J. Number Theory 124 (2007), no. 1, 193–199. MR2321000
Jun 12, 2020 at 1:54 comment added Lucia You may be interested to know that the result of Erdos, Stewart and Tijdeman was improved in work of Konyagin and Soundararajan (arxiv.org/abs/math/0604453; published paper in JNT), where the $\sqrt{s}$ is improved to $s^{2-\sqrt{2}-\epsilon}$.
Jun 12, 2020 at 1:47 history answered Joe Silverman CC BY-SA 4.0