Problems & Puzzles:
A question about consecutive integers
R. Devesh sent the following puzzle.
following pairs of consecutive numbers n
and n+1 have the property that the
largest prime factor of n and n+1 is
less than the ln(n) and ln(n+1)
2400, 2401 ==> (2,3,5), (7) ==> 5 < ln(2400)
and 7 < ln(2401)
4374, 4375 ==> (2,3), (5,7) ==> 3 < ln(4374) and
7 < ln(4375)
Q1: Find the next pair.
Q2: Are there any triplets?
Contributions came from Fred Schneider and Simon Cavegn
Simon Cavegn wrote:
I found no other consecutive pair (yet).
Searched all integers up 4400000000000000.
It's way faster to multiply prime powers than trying
Check up to 4400000000000000 took 8s, and found 16152018 numbers
with largest prime factor < ln(n)