Cubic Number

A Figurate Number of the form $n^3$, for $n$ a Positive Integer. The first few are 1, 8, 27, 64, ... (Sloane's A000578). The Generating Function giving the cubic numbers is

$${x(x^2+4x+1)\over(x-1)^4}=x+8x^2+27x^3+\ldots.$$ (1)

The Hex Pyramidal Numbers are equivalent to the cubic numbers (Conway and Guy 1996).

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, ...(Sloane's 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, ... (Sloane's A003108). In the early twentieth century, Dickson, Pillai, and Niven proved that every Positive Integer is the sum of not more than nine Cubes (so $g(3)=9$ in Waring's Problem).

In 1939, Dickson proved that the only Integers requiring nine Cubes are 23 and 239. Wieferich proved that only 15 Integers require eight Cubes: 15, 22, 50, 114, 167, 175, 186, 212, 213, 238, 303, 364, 420, 428, and 454 (Sloane's A018889). The quantity $G(3)$ in Waring's Problem therefore satisfies $G(3)\leq 7$, and the largest number known requiring seven cubes is 8042. The following table gives the first few numbers which require at least $N=1$, 2, 3, ..., 9 (positive) cubes to represent them as a sum.

$N$	Sloane	Numbers
1	Sloane's A000578	1, 8, 27, 64, 125, 216, 343, 512, ...
2	Sloane's A003325	2, 9, 16, 28, 35, 54, 65, 72, 91, ...
3	Sloane's A003072	3, 10, 17, 24, 29, 36, 43, 55, 62, ...
4	Sloane's A003327	4, 11, 18, 25, 30, 32, 37, 44, 51, ...
5	Sloane's A003328	5, 12, 19, 26, 31, 33, 38, 40, 45, ...
6	Sloane's A003329	6, 13, 20, 34, 39, 41, 46, 48, 53, ...
7	Sloane's A018890	7, 14, 21, 42, 47, 49, 61, 77, ...
8	Sloane's A018889	15, 22, 50, 114, 167, 175, 186, ...
9	Sloane's A018888	23, 239

There is a finite set of numbers which cannot be expressed as the sum of distinct cubes: 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, ...(Sloane's A001476). The following table gives the numbers which can be represented in exactly $W$ different ways as a sum of $N$ positive cubes. For example,

$$157=4^3+4^3+3^3+1^3+1^3=5^3+2^3+2^3+2^3+2^3$$ (2)

can be represented in $W=2$ ways by $N=5$ cubes. The smallest number representable in $W=2$ ways as a sum of $N=2$ cubes,

$$1729=1^3+12^3=9^3+10^3,$$ (3)

is called the Hardy-Ramanujan Number and has special significance in the history of mathematics as a result of a story told by Hardy about Ramanujan . Sloane's A001235 is defined as the sequence of numbers which are the sum of cubes in two or more ways, and so appears identical in the first few terms.

$N$	$W$	Sloane	Numbers
1	1	Sloane's A000578	1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ...
2	1		2, 9, 16, 28, 35, 54, 65, 72, 91, 126, ...
2	2		1729, 4104, 13832, 20683, 32832, 39312, ...
2	3	Sloane's A003825	87539319, 119824488, 143604279, 175959000, ...
2	4	Sloane's A003826	6963472309248, 12625136269928, 21131226514944, ...
2	5		48988659276962496, 490593422681271000, ...
2	6		8230545258248091551205888, ...
3	1	Sloane's A025395	3, 10, 17, 24, 29, 36, 43, 55, 62, 66, 73, ...

It is believed to be possible to express any number as a Sum of four (positive or negative) cubes, although this has not been proved for numbers of the form $9n\pm 4$. In fact, all numbers not of the form $9n\pm 4$ are known to be expressible as the Sum of three (positive or negative) cubes except 30, 33, 42, 52, 74, 110, 114, 156, 165, 195, 290, 318, 366, 390, 420, 435, 444, 452, 462, 478, 501, 530, 534, 564, 579, 588, 600, 606, 609, 618, 627, 633, 732, 735, 758, 767, 786, 789, 795, 830, 834, 861, 894, 903, 906, 912, 921, 933, 948, 964, 969, and 975 (Sloane's A046041; Guy 1994, p. 151).

The following table gives the possible residues (mod $n$) for cubic numbers for $n=1$ to 20, as well as the number of distinct residues $s(n)$.

$n$	$s(n)$	$x^3{\rm\ (mod\ }n)$
2	2	0, 1
3	3	0, 1, 2
4	3	0, 1, 3
5	5	0, 1, 2, 3, 4
6	6	0, 1, 2, 3, 4, 5
7	3	0, 1, 6
8	5	0, 1, 3, 5, 7
9	3	0, 1, 8
10	10	0, 1, 2, 3, 4, 5, 6, 7, 8, 9
11	11	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
12	9	0, 1, 3, 4, 5, 7, 8, 9, 11
13	5	0, 1, 5, 8, 12
14	6	0, 1, 6, 7, 8, 13
15	15	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
16	10	0, 1, 3, 5, 7, 8, 9, 11, 13, 15
17	17	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
18	6	0, 1, 8, 9, 10, 17
19	7	0, 1, 7, 8, 11, 12, 18
20	15	0, 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19

Dudeney found two Rational Numbers other than 1 and 2 whose cubes sum to 9,

$${415280564497\over 348671682660} {\rm\ and\ } {676702467503\over 348671682660}.$$ (4)

The problem of finding two Rational Numbers whose cubes sum to six was ``proved'' impossible by Legendre. However, Dudeney found the simple solutions 17/21 and 37/21.

The only three consecutive Integers whose cubes sum to a cube are given by the Diophantine Equation

$$3^3+4^3+5^3=6^3.$$ (5)

Catalan's Conjecture states that 8 and 9 ($2^3$ and $3^2$) are the only consecutive Powers (excluding 0 and 1), i.e., the only solution to Catalan's Diophantine Problem. This Conjecture has not yet been proved or refuted, although R. Tijdeman has proved that there can be only a finite number of exceptions should the Conjecture not hold. It is also known that 8 and 9 are the only consecutive cubic and Square Numbers (in either order).

There are six Positive Integers equal to the sum of the Digits of their cubes: 1, 8, 17, 18, 26, and 27 (Sloane's A046459; Moret Blanc 1879). There are four Positive Integers equal to the sums of the cubes of their digits:

$$153 = 1^3+5^3+3^3$$ (6)

$$370 = 3^3+7^3+0^3$$ (7)

$$371 = 3^3+7^3+1^3$$ (8)

$$407 = 4^3+0^3+7^3$$ (9)

(Ball and Coxeter 1987). There are two Square Numbers of the form $n^3-4$: $4=2^3-4$ and $121=5^3-4$ (Le Lionnais 1983). A cube cannot be the concatenation of two cubes, since if $c^3$ is the concatenation of $a^3$ and $b^3$, then $c^3 = 10^k a^3 + b^3$, where $k$ is the number of digits in $b^3$. After shifting any powers of 1000 in $10^k$ into $a^3$, the original problem is equivalent to finding a solution to one of the Diophantine Equations

$$c^3 - b^3 = a^3$$ (10)

$$c^3 - b^3 = 10 a^3$$ (11)

$$c^3 - b^3 = 100 a^3.$$ (12)

None of these have solutions in integers, as proved independently by Sylvester, Lucas, and Pepin (Dickson 1966, pp. 572-578).

See also Biquadratic Number, Centered Cube Number, Clark's Triangle, Diophantine Equation--Cubic, Hardy-Ramanujan Number, Partition, Square Number

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 14, 1987.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 42-44, 1996.

Davenport, H. ``On Waring's Problem for Cubes.'' Acta Math. 71, 123-143, 1939.

Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1966.

Guy, R. K. ``Sum of Four Cubes.'' §D5 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 151-152, 1994.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 53, 1983.

Sloane, N. J. A. Sequences A000578/M4499, A002376/M0466, and A003108/M0209 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.