Problems & Puzzles: Puzzles

Puzzle 901. A question about consecutive integers Abhiram R. Devesh sent the following puzzle. The 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) respectively eg.  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)   Questions   Q1: Find the next pair. Q2: Are there any triplets?

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Contributions came from Fred Schneider and Simon Cavegn

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Fred wrote:

The question is answered here http://oeis.org/A147302

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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 factorization.

Check up to 4400000000000000 took 8s, and found 16152018 numbers with largest prime factor < ln(n)

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Simon Cavegn wrote on Dec. 23,2017:

Here is an addition to puzzle 901:

 

There is no other pair than the two known ones, up to 6837671229762744147024034136064.
Tested this range with prime factors up to and including 67.
Due to computation time the first prime factor not tested was 71, and therefore the search was limited to about e^71 or about 10^30.8349
There were 1850494771855 (single / non-pair) numbers found in this range with largest prime factor < ln(n).
The search took 8d 9h and 12 parallel threads. (After optimizing the algorithm quite a bit in c#.)

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