Editing Powerful number (section)

== Sums and differences of powerful numbers ==
Any odd number is a difference of two consecutive squares: (''k'' + 1)<sup>2</sup> = ''k''<sup>2</sup> + 2''k'' + 1, so (''k'' + 1)<sup>2</sup>&nbsp;&minus;&nbsp;''k''<sup>2</sup> = 2''k'' + 1. Similarly, any multiple of four is a difference of the squares of two numbers that differ by two: (''k'' + 2)<sup>2</sup>&nbsp;&minus;&nbsp;''k''<sup>2</sup> = 4''k'' + 4. However, a [[singly even number]], that is, a number divisible by two but not by four, cannot be expressed as a difference of squares. This motivates the question of determining which singly even numbers can be expressed as differences of powerful numbers. Golomb exhibited some representations of this type:

:2 = 3<sup>3</sup>&nbsp;&minus;&nbsp;5<sup>2</sup>
:10 = 13<sup>3</sup>&nbsp;&minus;&nbsp;3<sup>7</sup>
:18 = 19<sup>2</sup>&nbsp;&minus;&nbsp;7<sup>3</sup> = 3<sup>5</sup>&nbsp;&minus;&nbsp;15<sup>2</sup>.

It had been conjectured that 6 cannot be so represented, and Golomb conjectured that there are infinitely many integers which cannot be represented as a difference between two powerful numbers. However, Narkiewicz showed that 6 can be so represented in infinitely many ways such as

:6 = 5<sup>4</sup>7<sup>3</sup>&nbsp;&minus;&nbsp;463<sup>2</sup>,

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

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