A332201 Sum of three cubes problem: a(n) = integer x with the least possible absolute value such that n = x^3 + y^3 + z^3 with |x| >= |y| >= |z|, or 0 if no such x exists.

0, 1, 1, 1, 0, 0, 2, 2, 2, 2, 2, 3, -11, 0, 0, 2, 2, 2, 3, 3, 3, 16, 0, 0, 2, 3, 3, 3, 3, 3, 2220422932, 0, 0

Offset: 0

Author

M. F. Hasler, Feb 08 2020

Keywords

• sign
• more
• hard

Comments

It is known that there is no solution for n congruent to +-4 (mod 9), but it is now conjectured that there is a solution (and probably infinitely many such) for all other numbers. The numbers n = 0, 1 and 2 are the only cases for which infinite families of parametric solutions are known, for other n the solutions seem to be sporadic.

Search on this problem was motivated by a statement in Mordell's paper from 1953. Beck et al. found a solution for n = 30 in 1999, and for 52 in 2000. Huisman found a solution for n = 74 in 2016. A solution for 33 was found by Booker in 2019. The number 42 was the last one below 100 for which a solution was found, in late 2019, using a collaborative effort with supercomputers and home computers from volunteers.

For n < 30, we have a(n) = A246869(n+1) for the nonzero values, while A246869(n+1) = 2 for n == 4 or 5 (mod 9) up to there.

Examples

			   0 = 0^3 + 0^3 + 0^3,     1 = 1^3 + 0^3 + 0^3,
   2 = 1^3 + 1^3 + 0^3,     3 = 1^3 + 1^3 + 1^3,
   6 = 2^3 - 1^3 - 1^3,     7 = 2^3 - 1^3 + 0^3,
   8 = 2^3 + 0^3 + 0^3,     9 = 2^3 + 1^3 + 0^3,
  10 = 2^3 + 1^3 + 1^3,    11 = 3^3 - 2^3 - 2^3,
  12 = -11^3 + 10^3 + 7^3, 15 = 2^3 + 2^3 - 1^3,
  16 = 2^3 + 2^3 + 0^3,    17 = 2^3 + 2^3 + 1^3,
  18 = 3^3 - 2^3 - 1^3,    19 = 3^3 - 2^3 + 0^3,
  20 = 3^3 - 2^3 + 1^3,    21 = 16^3 - 14^3 - 11^3,
  24 = 2^3 + 2^3 + 2^3,    25 = 3^3 - 1^3 - 1^3,
  26 = 3^3 - 1^3 + 0^3,    27 = 3^3 + 0^3 + 0^3,
  28 = 3^3 + 1^3 + 0^3,    29 = 3^3 + 1^3 + 1^3,
30 = 2220422932^3 - 2218888517^3 - 283059965^3 was discovered by Beck, Pine, Yarbrough and Tarrant in 1999 following an approach suggested by N. Elkies.
33 = 8866128975287528^3 - 8778405442862239^3 - 2736111468807040^3 was found by A. Booker in 2019. It is uncertain whether these are the smallest solutions.
		

Links

• Michael Beck, Eric Pine, Wayne Tarrant and Kim Yarbrough Jensen, New integer representations as the sum of three cubes, Math. Comp. 76 (2007) pp. 1683-1690, DOI:10.1090/S0025-5718-07-01947-3.
• A. R. Booker, Cracking the problem with 33, Res. Number Theory 5 no. 26 (2019), DOI:10.1007/s40993-019-0162-1.
• V. L. Gardiner, R. B. Lazarus and P. R. Stein: Solutions of the Diophantine Equation x^3 + y^3 = z^3 - d, Math.Comp. 18, No. 87 (1964), pp. 408-413. DOI: 10.2307/2003763.
• Jon Grantham and P. G. Walsh, Representing integers as a sum of three cubes, arXiv preprint, arXiv:2211.12149 [math.NT], 2022.
• B. Haran, Sum of Three Cubes - Numberphile, YouTube playlist (featuring videos from Nov 06 2015, May 31 2016 (about 74), Mar 12 2019 (about 33), Sep 2019 (about 42)).
• Sander G. Huisman, Newer sums of three cubes, arXiv:1604.07746 [math.NT] (2016).
• J. C. P. Miller and M. F. C. Woollett, Solutions of the Diophantine Equation x^3+y^3+z^3=k, Journal of the London Mathematical Society, s1-30 (1955) pp. 101-110. DOI: 10.1112/jlms/s1-30.1.101.
• L. J. Mordell, Integer Solutions of the Equation x^2+y^2+z^2+2xyz = n, Journal London Math. Soc. s1-28 no. 4 (1953) pp. 500-510.
• Bjorn Poonen, Undecidability in number theory, Notices Amer. Math. Soc. 55 (2008), no. 3, pp. 344-350.

Crossrefs

Cf. A060464, A060465, A060466, A060467, A246869.

Programs

PARI

apply( A332201(n,L=oo)={!bittest(48,n%9)&& for(c=0,L, my(t1=c^3-n, t2=c^3+n, a); for(b=0,c,((ispower(t1-b^3,3,&a)&&abs(a)<=c)||(ispower(t1+b^3,3,&a)&&abs(a)<=c))&&return(c); ispower(t2-b^3,3,&a) && abs(a)<=c && return(-c)))}, [0..29])

Formula

a(n) = 0 for n == 4 or n == 5 (mod 9).

a(n) <= k if |n - k^3| < 3 or |n - 2*k^3| < 2 or n = 3*k^3 for some k.

a(n) = A246869(n+1) for all n < 30 with a(n) > 0.

Extensions

a(31) = a(32) = 0 added by Jinyuan Wang, Feb 15 2020