Powerful number: Difference between revisions

Revision as of 03:15, 12 September 2006

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A '''powerful number''' is a [[negative and non-negative numbers|positive]] [[integer]] ''m'' that for every [[prime number|prime]] number ''p'' dividing ''m'', ''p''² also divides ''m''. Equivalently, a powerful number is the [[Product (mathematics)|product]] of a [[Square number|square]] and a [[Cube (arithmetic)|cube]], that is, a number ''m'' of the form ''m'' = ''a''²''b''³. Powerful numbers are also known as '''squareful''', '''square-full''', or '''2-full'''. [[Paul Erdős]] and [[George Szekeres]] studied such numbers and [[Solomon W. Golomb|Golomb]] named such numbers powerful.

The following is a list of all powerful numbers between 1 and 1000:

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(Golomb, 1970).

The two smallest consecutive powerful numbers are 8 and 9. Since [[Pell's equation]] ''x''² − 8''y''² = 1 has infinitely many integral solutions, there are infinitely many pairs of consecutive powerful numbers (Golomb, 1970). The sequence of pairs of consecutive numbers is given by {{OEIS|id=A060355}}. It is a [[Erdős conjecture|conjecture]] of [[Paul Erdős|Erdős]], Mollin, and Walsh that there are no three consecutive powerful numbers; if the [[abc conjecture]] is true, it would follow that there are only finitely many triples of consecutive powerful numbers.

== Sums and differences of powerful numbers ==

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and McDaniel showed that every integer has infinitely many such representations(McDaniel, 1982).

[[Erdős]] conjectured that every sufficiently large integer is a sum of at most three powerful numbers; this was proved by [[Roger Heath-Brown]] (Heath-Brown, 1987).

== Generalization ==