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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

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.

LINKS

Table of n, a(n) for n=0..32.

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.

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, 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.

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.

EXAMPLE

   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.

PROG

(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])

CROSSREFS

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

Sequence in context: A029116 A064770 A322073 * A060467 A125918 A239202

Adjacent sequences:  A332196 A332197 A332200 * A332202 A332203 A332204

KEYWORD

sign,more,hard

AUTHOR

M. F. Hasler, Feb 08 2020

EXTENSIONS

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

STATUS

approved

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Last modified July 6 23:56 EDT 2020. Contains 335484 sequences. (Running on oeis4.)