Powerful numbers (4,8,9,16,25,27... )are defined as follows: if a prime p divides n then p^{2} must also divide n.
(8,9) is a couple of two consecutive numbers,both of them being powerful.
Find another pair(s) like that.
The more the merrier!!
The first five powerful pairs are:
(8,9), (288, 289), (675, 676), (9800, 9801) and (12167, 12168).
Related to powerful numbers are Achilles numbers. Achilles numbers are powerful numbers that are not perfect powers, such as one number of each powerful pair given of the latter four: 288 [2^{5} × 3^{2}], 675 [3^{3} × 5^{2}], 9800 [2^{3} × 5^{2} × 7^{2}], and 12168 [2^{3} × 3^{2} × 13^{2}].
The first Achilles pair (and therefore another powerful pair) is (5425069447, 5425069448): 5425069447 [7^{3} × 41^{2} × 97^{2}] ; 5425069448 [2^{3} × 26041^{2}] .
A perfect power is where the integer can be expressed as m^{k} such that m and k are integers > 1.
To be more precise in definition of a powerful number, a powerful number is a positive integer m where every prime factor p dividing m, p^{2 }also divides m.
Edited on August 9, 2010, 7:23 pm

Posted by Dej Mar
on 20100809 14:20:01 