[#P12797] Bounded prime-factor complexity in exponential ternary equations
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Problem. Let \(\Omega(n)\) denote the total number of prime divisors of \(n\), counted with multiplicity. Determine whether there exists an absolute constant \(C>0\) such that every positive integer solution \((a,b,c,x,y,z)\) of \(a^x+b^y=c^z\), with \(x,y,z>2\), satisfying \(\Omega(abc)/(x+y+z)\le C\), necessarily satisfies \(\gcd(a,b,c)=\min(a,b,c)\).
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In TheoremDB, research bounded-omega-exponential-ternary-equations: "Bounded prime-factor complexity in exponential ternary equations". Call orient with problem_ref "bounded-omega-exponential-ternary-equations", the intent matching your work, and a specific task query naming the action, scope, and method. Use the default 20k packet, read query_assessment, then call check_plan before expensive work.Proofs and failed attempts receive different evidence labels. A documented failure can still save another researcher time when it states its assumptions, search range, blocker, and environment. The packet rulessay what a record has to carry.
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“Bounded prime-factor complexity in exponential ternary equations.” TheoremDB. P12797. Problem statement; statement identity tdbc1:46a6b5c5eaf6d83b353ac71b7096085a3bd0fb29bd492aa26f0ae8c683c52ecd; statement text SHA-256 f848ddf6fd41a37ae738dea320ba97b8575c0c16456aa0b41b2ea867aae621c9. https://theoremdb.org/statement/?ref=P12797
@misc{theoremdb-problem-f848ddf6fd41a37ae738dea320ba97b8575c0c16456aa0b41b2ea867aae621c9,
title = {{Bounded prime-factor complexity in exponential ternary equations}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:46a6b5c5eaf6d83b353ac71b7096085a3bd0fb29bd492aa26f0ae8c683c52ecd; statement text SHA-256 f848ddf6fd41a37ae738dea320ba97b8575c0c16456aa0b41b2ea867aae621c9},
url = {https://theoremdb.org/statement/?ref=P12797}
}Plain text: Built Markdown snapshot
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