Powerful numbers (4,8,9,16,25,27... )are defined as follows: if a prime p divides n then p² 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!!

Pairs are very common, unless I am misreading the puzzle, and not at all difficut to find.  Up to 1000:

(8,9) (24,25) (27,28) (44,45) (48,49.50) (63,64)  (75,76) (80,81) (98,99,100) (116,117) (120,121) (124,125), (135,136) (147,148) (152,153) (168,169) (171-172) (175,176) (188,189) (207,208) (224,225) (243,244,245) (260,261) (275,276) (279,280) (288,289) (296,297) (315,316) (324,325) (332,333) (342,343,344) (350,351,352) (360,361) (363,364) (368,369) (375,376) (387,388) (404,405), (423,424,425) (440,441) (459,460) (475,476,477) (495,496) (507,508) (512,513), (524,525) (528,529) (531,532) (539,540) (548,549,550) (567,568) (575,576) (584-585) (603,604,605) (620,621) (624,625) (636-637) (639,640) (656,657) (675,676) (692,693) (711,712) (724,725,726) (728,729) (735,736) (764,765) (774,775,776) (783,784) (800,801) (819,820) (824,825) (832,833) (836,837) (840,841), (844,845,846,847,848) (855,856) (867,868) (872,873) (875,876) (891-892) (908,909) (924,925) (927,928) (931,932) (944,945) (960,961) (963,964) (975,976) (980,981)

These are all divisible by the square of a prime. There are not only plenty of pairs, but also multiples -- including one of five consecutive numbers (844..848 -- see above).  With this much typing, there are probably some typos, but I'll wait to see if the problem is clarified.

Posted by ed bottemiller on 2010-08-09 13:25:01