Powerful number: Difference between revisions

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Any odd number is a difference of two consecutive squares: 2''k'' + 1 = (''k'' + 1)² - ''k''². Similarly, any multiple of four is a difference of the squares of two numbers that differ by two. 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 = 53²³ − 35³²

:10 = 13³ − 3⁷

:18 = 19² − 7³ = 3²(3³ − 5²).