PowersRepresentations

The ways to write $n$ as a sum of $k$ non-negative $p$ th powers, as non-decreasing lists.

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PowersRepresentations(n, k, p)every non-decreasing

(x_1 \le \dots \le x_k)

with

\sum x_i^p = n

Typed signatures

Type	Package	Applies
`(integer, integer, integer) -> list`	`number-theory`	always

Domain: Number theory

Details

Examples

Basic #

\mathrm{PowersRepresentations}(25, 2, 2)

Out

\bigl\lbrack\bigl\lbrack0, 5\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack

The smallest number that is a sum of two positive squares in two ways

\mathrm{PowersRepresentations}(50, 2, 2)

Out

\bigl\lbrack\bigl\lbrack1, 7\bigr\rbrack, \bigl\lbrack5, 5\bigr\rbrack\bigr\rbrack

Scope #

\mathrm{PowersRepresentations}(100, 3, 2)

Out

\bigl\lbrack\bigl\lbrack0, 0, 10\bigr\rbrack, \bigl\lbrack0, 6, 8\bigr\rbrack\bigr\rbrack

Ways to write 87,539,319 as a sum of two cubes

\mathrm{PowersRepresentations}(87\,539\,319, 2, 3)

Out

\bigl\lbrack\bigl\lbrack167, 436\bigr\rbrack, \bigl\lbrack228, 423\bigr\rbrack, \bigl\lbrack255, 414\bigr\rbrack\bigr\rbrack

Properties #

Sums of three squares, up to ordering

\mathrm{PowersRepresentations}(102, 3, 2)

Out

\bigl\lbrack\bigl\lbrack1, 1, 10\bigr\rbrack, \bigl\lbrack2, 7, 7\bigr\rbrack\bigr\rbrack

\mathrm{Filter}(\mathrm{PowersRepresentations}(7, 3, 1), ()↦\mathrm{FreeQ}(\operatorname{\_1}, 0))

Out

\bigl\lbrack\bigl\lbrack1, 1, 5\bigr\rbrack, \bigl\lbrack1, 2, 4\bigr\rbrack, \bigl\lbrack1, 3, 3\bigr\rbrack, \bigl\lbrack2, 2, 3\bigr\rbrack\bigr\rbrack

Neat examples #

The taxicab number $1729=1^3+12^3=9^3+10^3$

\mathrm{PowersRepresentations}(1\,729, 2, 3)

Out

\bigl\lbrack\bigl\lbrack1, 12\bigr\rbrack, \bigl\lbrack9, 10\bigr\rbrack\bigr\rbrack

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