A cubic number is a figurate number of the form 
with 
a positive
integer. The first few are 1, 8, 27, 64, 125, 216, 343, ... (OEIS A000578).
The protagonist Christopher in the novel The
Curious Incident of the Dog in the Night-Time recites the cubic numbers to
calm himself and prevent himself from wanting to hit someone (Haddon 2003, p. 213).
The generating function giving the cubic numbers is
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(1)
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The hex pyramidal numbers are equivalent to the cubic numbers (Conway and Guy 1996).
The plots above show the first 255 (top figure) and 511 (bottom figure) cubic numbers represented in binary.
Pollock (1843-1850) conjectured that every number is the sum of at most 9 cubic numbers (Dickson 2005, p. 23). As a part of the study of Waring's
problem, it is known that every positive integer is a sum of no more than 9 positive
cubes (
,
proved by Dickson, Pillai, and Niven in the early twentieth century), that every
"sufficiently large" integer is a sum of no more than 7 positive cubes
(
). However, it is not known
if 7 can be reduced (Wells 1986, p. 70). The number of positive cubes
needed to represent the numbers 1, 2, 3, ... are 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4,
5, 6, 7, 8, 2, ...(OEIS A002376), and the number
of distinct ways to represent the numbers 1, 2, 3, ... in terms of positive cubes
are 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4,
5, 5, 5, 5, ... (OEIS A003108).
In 1939, Dickson proved that the only integers requiring nine positive cubes are 23 and 239. Wieferich proved that only 15 integers
require eight cubes: 15, 22, 50, 114, 167, 175, 186, 212, 231, 238, 303, 364, 420,
428, and 454 (OEIS A018889). The quantity 
in Waring's
problem therefore satisfies 
, and the largest number known requiring seven cubes
is 8042. Deshouillers et al. (2000) conjectured that 7373170279850 is the
largest integer that cannot be expressed as the sum of four nonnegative cubes.
The following table gives the first few numbers which require at least 
, 2, 3, ..., 9 (i.e., 
or more) positive cubes to represent them as a sum.
 | OEIS | numbers |
| 1 | A000578 | 1, 8, 27, 64, 125, 216, 343, 512, ... |
| 2 | A003325 | 2, 9, 16, 28, 35, 54, 65, 72, 91, ... |
| 3 | A047702 | 3, 10, 17, 24, 29, 36, 43, 55, 62, ... |
| 4 | A047703 | 4, 11, 18, 25, 30, 32, 37, 44, 51, ... |
| 5 | A047704 | 5, 12, 19, 26, 31, 33, 38, 40, 45, ... |
| 6 | A046040 | 6, 13, 20, 34, 39, 41, 46, 48, 53, ... |
| 7 | A018890 | 7, 14, 21, 42, 47, 49, 61, 77, ... |
| 8 | A018889 | 15, 22, 50, 114, 167, 175, 186, ... |
| 9 | A018888 | 23, 239 |
There is a finite set of numbers which cannot be expressed as the sum of distinct positive cubes: 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, ...(OEIS A001476).
It is known that every integer is a sum of at most 5 signed cubes (
in Waring's problem).
It is believed that 5 can be reduced to 4, so that
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(2)
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for any number 
,
although this has not been proved for numbers of the form 
. However, every multiple of 6 can
be represented as a sum of four signed cubes as
a result of the algebraic identity
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(3)
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In fact, all numbers 
and not of the form 
are known to be expressible as the sum
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(4)
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of three (positive or negative) cubes with the exception of 
, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, and
975 (Miller and Woollett 1955; Gardiner et al. 1964; Guy 1994, p. 151;
Mishima; Elsenhaus and Jahnel 2007; Booker; Huisman 2016). Examples include:
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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