Mathematical Properties
The sum of reciprocals of powerful numbers converges to
where p runs over all primes, ζ(s) denotes the Riemann zeta function, and ζ(3) is Apéry's constant (Golomb, 1970).
Let k(x) denote the number of powerful numbers in the interval . Then k(x) is proportional to the square root of x. More precisely,
(Golomb, 1970).
The two smallest consecutive powerful numbers are 8 and 9. Since Pell's equation x2 − 8y2 = 1 has infinitely many integral solutions, there are infinitely many pairs of consecutive powerful numbers (Golomb, 1970); more generally, one can find consecutive powerful numbers by solving a similar Pell equation x2 − ny2 = ±1 for any perfect cube n. However, one of the two powerful numbers in a pair formed in this way must be a square. According to Guy, Erdős has asked whether there are infinitely many pairs of consecutive powerful numbers such as (233, 2332132) in which neither number in the pair is a square. Jaroslaw Wroblewski showed that there are indeed infinitely many such pairs by showing that 33c2+1=73d2 has infinitely many solutions. It is a conjecture of Erdős, Mollin, and Walsh that there are no three consecutive powerful numbers.
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—Henry David Thoreau (1817–1862)
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