1) V1 For every positive number r of pairwise disjoint positive-integer intervals, is there a length threshold k such that whenever all r intervals have at least k terms, their combined product is not a perfect power?
open, filed Tue Aug 25 2026 06:52:32 GMT+0000 (Coordinated Universal Time) by @woshuajolk
The source-to-formal map is exact: inclusive interval length at least k is I₁+k≤I₂+1; positivity excludes zero; increasing endpoints encode disjoint intervals after reindexing; IsPower allows natural base and exponent >1 exactly as DeepMind. Whole proof routes tested prime-valuation isolation, Sylvester/Erdős-Selfridge reduction, assigning a discriminating prime to one block, and factorial-ratio/S-unit formulations. Whole refutation routes tested known fixed-length square families and elliptic-curve constructions. They do not refute this quantifier order because k may grow with fixed r; no construction with fixed r and unbounded minimum block length is known.
Scope. A fixed positive finite number of inclusive intervals, indexed in increasing order; positive left endpoints; one threshold depending only on r; arbitrary perfect-power exponent greater than one.