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A060465 Value of x of the solution to x^3 + y^3 + z^3 = A060464(n) (numbers not 4 or 5 mod 9) with smallest |z| and smallest |y|, 0 <= |x| <= |y| <= |z|. 4
0, 0, 0, 1, -1, 0, 0, 0, 1, -2, 7, -1, -511, 1, -1, 0, 1, -11, -2901096694, -1, 0, 0, 0, 1, -283059965 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

Indexed by A060464.

Only primitive solutions where gcd(x,y,z) does not divide n are considered.

From the solution A060464(24) = 30 = -283059965^3 - 2218888517^3 + 2220422932^3 (smallest possible magnitudes according to A. Bogomolny), one has a(24) = -283059965. A solution to A060464(25) = 33 remains to be found. Other values for larger n can be found in the first column of the table on Hisanori Mishima's web page. - M. F. Hasler, Nov 10 2015

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, New York, 2004, Section D5, 231-234.

LINKS

Table of n, a(n) for n=0..24.

D. J. Bernstein, Three cubes

A. Bogomolny, Finicky Diophantine Equations on cut-the-knot.org, accessed Nov. 10, 2015

B. Conn, L. Vaserstein, On sums of three integral cubes, Contemp. Math 166 (1994) MR1284068

V. L. Gardiner, R. B. Lazarus, P. R. Stein, Solutions of the diophantine equation x^3+y^3=z^3-d, Math. Comnp. 18 (1964) 408-413

D.R. Heath-Brown, W.M. Lioen and H.J.J. te Riele on Solving the Diophantine Equation x3 + y3 + z3 = k on a Vector Computer

J. C. P. Miller, M. F. C. Woollett, Solutions of the Diophantine Equation x^3+y^3+z^3=k, J. Lond. Math. Soc. 30 (1) (1955) 101-110

Hisanori Mishima, About n=x^3+y^3+z^3

EXAMPLE

For n=16 the smallest solution is 16 = (-511)^3 + (-1609)^3 + 1626^3, which gives the term -511.

MATHEMATICA

(* this program is not convenient for hard cases *) nmax = 29; xmin[_] = 0; xmax[_] = 20; xmin[16] = 500; xmax[16] = 600; xmin[24] = 2901096600; xmax[24] = 2901096700; r[n_, x_] := Reduce[0 <= Abs[x] <= Abs[y] <= Abs[z] && n == x^3 + y^3 + z^3, {y, z}, Integers]; r[n_ /; IntegerQ[n^(1/3)]] := {0, 0, n^(1/3)}; mySort = Sort[#1, Which[Abs[#1[[3]]] <= Abs[#2[[3]]], True, Abs[#1[[3]]] == Abs[#2[[3]]], If[Abs[#1[[2]]] <= Abs[#2[[2]]], True, False], True, False] & ] & ; rep := {x_, y_, z_} /; (x + y == 0 && x > 0) :> {-x, -y, z}; r[n_] := Reap[Do[ sp = r[n, x] /. C[1] -> 1; If[sp =!= False, xyz = {x, y, z} /. {ToRules[sp]} /. rep; If[GCD @@ Flatten[{n, xyz}] == 1, Sow[xyz]]]; sn = r[n, -x] /. C[1] -> 1; If[sn =!= False, xyz = {-x, y, z} /. {ToRules[sn]} /. rep; If[GCD @@ Flatten[{n, xyz}] == 1, Sow[xyz]]], {x, xmin[n], xmax[n]}]][[2, 1]] // Flatten[#, 1] & // mySort // First; A060464 = Select[Range[0, nmax], Mod[#, 9] != 4 && Mod[#, 9] != 5 &]; A060465 = Table[xyz = r[n]; Print[ " n = ", n, " {x, y, z} = ", xyz]; xyz[[1]], {n, A060464}] (* Jean-Fran├žois Alcover, Jul 10 2012 *)

CROSSREFS

Cf. A060466-A060467, A173515.

Sequence in context: A248676 A125699 A242207 * A219177 A139339 A090986

Adjacent sequences:  A060462 A060463 A060464 * A060466 A060467 A060468

KEYWORD

sign,nice,hard,more

AUTHOR

N. J. A. Sloane, Apr 10 2001

EXTENSIONS

Edited and a(24) added by M. F. Hasler, Nov 10 2015

STATUS

approved

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Last modified August 21 00:45 EDT 2017. Contains 290855 sequences.