
Glossary: Prime Pages: Top 5000: 
A positive integer n is powerful if for every prime p
dividing n, p^{2} also divides n. The reader
might want to pause and show that the powerful numbers are exactly those that can
be written a^{2}b^{3}, where a and b
are positive integers. Here are the powerful numbers
up to 1000:
1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, and 1000. There are pairs of consecutive powerful numbers such as: (8,9), (288,289), (675,676), (9800,9801), (12167,12168), (235224,235225), (332928,332929) and (465124,465125).Erdös conjectured in 1975 that there do not exist three consecutive powerful integers. Golomb also considered this question in 1970, as did Mollin and Walsh (independently) in 1986. The latter proved that the following are equivalent:
References:
Chris K. Caldwell © 19992016 (all rights reserved)
