The numbers that can be represented by a special cubic polynomials
Answers
Answer:
CubicNumber
A cubic number is a figurate number of the form n^3 with n 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
(x(x^2+4x+1))/((x-1)^4)=x+8x^2+27x^3+....
(1)
The hex pyramidal numbers are equivalent to the cubic numbers (Conway and Guy 1996).
Binary plot of the cubic numbers
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 (g(3)=9, 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 (G(3)<=7). 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 G(3) in Waring's problem therefore satisfies G(3)<=7, 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 N=1, 2, 3, ..., 9 (i.e., N or more) positive cubes to represent them as a sum.
N Sloane 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
Answer:
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.