# On partitions into four cubes

On partitions into four cubes
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We use partially ordered sets posets and graphs in order to obtain a formula for the number of partitions of a positive integer n into four cubes with two of them equal.

Tipo de documento: Artículo - Article

Palabras clave: compositions, cubic numbers, graphs, partitions, paths, posets.

Source: http://www.bdigital.unal.edu.co

## Teaser

Bol.

Mat.

16(2), 125–147 (2009)
125
On partitions into four cubes
Agustı́n Moreno1
Paola Palma2
Departamento de Matemáticas
Universidad Nacional de Colombia
Bogotá
Usamos conjuntos parcialmente ordenados (posets) y grafos para obtener
una fórmula para el número de particiones de un entero positivo n en
cuatro cubos con dos de ellos iguales.
Palabras Clave: composiciones, números cúbicos,
grafos, particiones, trayectorias, posets.
We use partially ordered sets (posets) and graphs in order to obtain a
formula for the number of partitions of a positive integer n into four
cubes with two of them equal.
Keywords: compositions, cubic numbers,
graphs, partitions, paths, posets.
MSC: 05A17, 11D45, 11D85, 11E25, 11P83, 16G20.
1
Introducción
We consider that part of Waring’s problem regarding cubes.

We must
recall that Waring, in his book Meditationes Algebraicae, published in
1770, stated without proof that every nonnegative integer is the sum of
four squares, nine cubes, 19 fourth powers and so on [19].
Waring’s problem for cubes is to prove that every nonnegative integer
is the sum of a finite number of nonnegative cubes.

The minimum such
number is denoted g(3).

Wieferevich and Kempner proved that g(3) = 9
[14].

This is clearly best possible, since there are integers, such as 23 and
239, that cannot be written as sum of eight cubes.
1
2
amorenoca@unal.edu.co
nppalmav@unal.edu.co
126
Moreno y Palma, On partitions into four cubes
Immediately after Wieferevich published his theorem, Landau observed that, in fact, only finitely many positive integers actually require
nine cubes, that is, every sufficiently large integer is the sum of eight
cubes, with the only exceptions being 23 and 239.
Linnik proved that every sufficiently large integer is a sum of 7 cubes;
Watson simplified the proof and McCurley gave an effective and explicit
proof of this result [17, 18, 21].
Demjanenko [7] proved that every number n 6≡ ±4 mod 9 can be expressed as t...