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powerful number




Also known as a squarefull number, a positive whole number n such that for every prime number p dividing n, p2 also divides n. Every powerful number can be written as a2b3, where a and b are positive integers. The first 20 powerful numbers 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, and 169. There are pairs of consecutive powerful numbers, such as (8,9), (288,289), and (675,676). However, no three consecutive powerful numbers are known and, in 1978, Paul Erdös conjectured that none exist.


Related category

   • TYPES OF NUMBERS