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

(1) |

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 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 in Waring's Problem therefore satisfies , and the largest number known requiring seven cubes is 8042. The following table gives the first few numbers which require at least , 2, 3, ..., 9 (positive) cubes to represent them as a sum.

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* different ways as a sum of positive cubes. For example,

(2) |

(3) |

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 . In fact, all numbers *not* of the form 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 ) for cubic numbers for to 20, as well as the number of distinct residues .

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,

(4) |

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

(5) |

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:

(6) | |||

(7) | |||

(8) | |||

(9) |

(Ball and Coxeter 1987). There are two Square Numbers of the form : and (Le Lionnais 1983). A cube cannot be the concatenation of two cubes, since if is the concatenation of and , then , where is the number of digits in . After shifting any powers of 1000 in into , the original problem is equivalent to finding a solution to one of the Diophantine Equations

(10) | |||

(11) | |||

(12) |

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

**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.

© 1996-9

1999-05-25