Cubic Number

A cubic number is a figurate number of the form with a nonnegative integer. The first few are 0, 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

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) positive 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 whose minimum number of positive cubes in a sum representing them is , 2, 3, ..., 9. Thus, each number in row is a sum of positive cubes but cannot be expressed using fewer.

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

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

In the case of three signed cubes, one seeks integer solutions of

Booker (2019) found representations for 33 and 795, and Booker and Sutherland (2021) found representations for 42, 165, 579, and 906 and a new representation for 3. Thus every positive not of the form was known to be a sum of three signed cubes by 2021, with the exceptions 114, 390, 627, 633, 732, 921, and 975, for which no representations were then known (Booker and Sutherland 2021). Examples from these and earlier searches (Miller and Woollett 1955; Gardiner et al. 1964; Guy 1994, p. 151; Elsenhans and Jahnel 2007; Huisman 2016; Mishima) include:

Equation (4) has no integer solutions for of the form , since a cube is congruent to 0, 1, or modulo 9, and three such residues cannot sum to 4 or 5 modulo 9 (Hardy and Wright 1979, p. 327). The seven exceptions listed above do not have this obstruction. Mahler proved that 1 has infinitely many representations as three signed cubes.

If one also excludes numbers of the form , every number can be represented as a sum of four signed cubes, using one of the following algebraic identities, or their complementary identities (via ):

These identities,and a proof for were given by Demjanenko (Demjanenko 1966, Cohen 2004).

The following table gives the numbers which can be represented in exactly different ways as a sum of positive cubes, with permutations of the summands counted only once. Taking the union over all for a given gives the numbers representable as sums of exactly positive cubes. This can include numbers whose minimum number of cubes is smaller than , such as . To obtain row of the previous table, also exclude numbers representable using fewer than positive cubes. For example,

can be represented in ways by cubes. The smallest number representable in ways as a sum of cubes,

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. Note that OEIS 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 to the series given below.

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

Dudeney found two rational numbers other than 1 and 2 whose cubes sum to nine,

(Gardner 1958). 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

Catalan's conjecture, proved by Mihăilescu (2004), states that 8 and 9 ( and ) are the only consecutive perfect powers greater than 1, i.e., the only solution to Catalan's Diophantine problem. In particular, 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 (OEIS A046459; Moret Blanc 1879). There are four positive integers equal to the sums of the cubes of their digits:

(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

None of these have solutions in integers, as proved independently by Sylvester, Lucas, and Pepin (Dickson 2005, pp. 572-578).