In this case, there exists the parametrized solution.
(x+1)3 + (x-1)3 = 2x3 + 6x
So
6x = (x+1)3 + (x-1)3 - 2x3
n | x | y | z |
---|---|---|---|
6 | 0 | 2 | -1 |
12 | 1 | 3 | -2 |
18 | 2 | 4 | -3 |
24 | 3 | 5 | -4 |
30 | 4 | 6 | -5 |
... | ... | ... | ... |
It is generally assumed 0 ≤ |x| ≤ |y| ≤ |z|.
Signs of x, y, z and coefficient 2 are following 24 cases,
(1) x, y, 2z |
(9) x, 2y, z |
(17) 2x, y, z |
(lexical order of z, y, x). In these cases,
(5) = -(4) |
(13) = -(12) |
(21) = -(20) |
so it is sufficient to search only 12 cases,
(1) (2) (3) (4) (9) (10) (11) (12) (17) (18) (19) (20)
In these cases, (1) (9) (17) are always positive for any x, y, z.
And (2) (3) (4) (10) (18) (19) are also always positive.
10 ' cube3.ub 20 M%=1000 : dim T%(M%) : for I%=1 to M% : T%(I%)=0 : next I% 30 ' 40 for Z%=0 to 10 50 for Y%=0 to Z% 60 for X%=0 to Y% 70 R%=fnSub(X% , Y% , Z% ,"z") : ' (1) 80 R%=fnSub(-X% , Y% , Z% ,"z") : ' (2) 90 R%=fnSub(X% , -Y% , Z% ,"z") : ' (3) 100 R%=fnSub(-X% , -Y% , Z% ,"z") : ' (4) 110 R%=fnSub(X% , Y% , Z% ,"y") : ' (9) 120 R%=fnSub(-X% , Y% , Z% ,"y") : ' (10) 130 R%=fnSub(X% , Y% , Z% ,"z") : ' (17) 140 R%=fnSub(-X% , Y% , Z% ,"z") : ' (18) 150 R%=fnSub(X% , -Y% , Z% ,"z") : ' (19) 160 next X% 170 next Y% 180 next Z% 190 end 200 ' 210 fnSub(X% , Y% , Z% , C) 220 local N , S% 230 N=X%^3+Y%^3+Z%^3 240 if C="x" then N=N+X%^3 250 if C="y" then N=N+Y%^3 260 if C="z" then N=N+Z%^3 270 if abs(N)>M% then 300 280 if T%(abs(N))=1 then 300 290 S%=sgn(N) : print S%*N , S%*X% , S%*Y% , S%*Z% , C : T%(S%*N)=1 300 return(0)
First we consider the case (11).
The range of y when z is fixed.
-1000 ≤ x3 - 2y3 + z3 ≤ 1000
z3 - 1000 ≤ - x3 + 2y3 ≤ z3 + 1000
Lower bound of y.
- x3 + 2y3 ≤ 2y3
so
z3 - 1000 ≤ 2y3
cbrt((z3-1000)/2) ≤ y
changes into integer,
int(cbrt((z3-1000)/2))+1 ≤ y
Upper bound of y.
- x3 + 2y3 ≥ y3
so
y3 ≤ z3 + 1000
y ≤ cbrt(z3+1000)
changes into integer,
y ≤ int(cbrt(z3+1000))
If z is sufficient large, the right side becomes almost equal to y.
Therefore upper bound of y can be regarded as z-1.
Next, decide the range of x when z and y are fixed.
-1000 ≤ x3 - 2y3 + z3 ≤ 1000
- z3+2y3-1000 ≤ x3 ≤ - z3+2y3+1000
so lower bound is
max{0, int(cbrt(-z3+2y3-1000))+1} ≤ x
Upper bound is
x ≤ int(cbrt(-z3+2y3+1000))
(12) -x, -2y, z
(20) -2x, -y, z
are the same ways.
10 ' cube4.ub : n = x^3 - 2y^3 + z^3 20 M%=1000:dim T%(M%):for I%=1 to M%:T%(I%)=0:next I% 30 ' 40 for Z=10 to 100000:Z3=Z^3 50 Ys=int(exp(log((Z3-M%)/2)/3))+1 60 for Y=Ys to Z-1:W=Z3-2*Y^3 70 Xs=-M%-W:if Xs≤0 then Xs=1 else Xs=int(exp(log(Xs)/3))+1 80 Xe=M%-W:Xe=int(exp(log(Xe)/3))+1 90 for X=Xs to Xe:gosub 140:next X 100 next Y 110 next Z 120 end 130 ' 140 N=W+X^3:V=abs(N):S=sgn(N):if V>M% then return 150 if T%(V)=1 then return 160 print V;":";X*S;",";-Y*S;",";Z*S;": y":T%(V)=1:return
I found the following new results,
230 = (-14101)3 + 272933 + 2(-20617)3
418 = 159613 + 917053 + 2(-72914)3
482 = (-2254)3 + (-11878)3 + 2(9449)3
580 = 851113 + 898453 + 2(-87542)3
(August 7, 2000)
Results by Soichirou Ichida,
76 = (-21167)3 + (-122171)3 + 2*971353
967 = 3806983 + 6412633 + 2*(-542246)3
And by Jean-Charles Meyrignac,
482 = (-2254)3 + (-11878)3 + 2*94493
76 = (-21167)3 + (-122171)3 + 2*971353
967 = 3806983 + 6412633 + 2*(-542246)3
356 = 1295213 + 10484693 + 2*(-832693)3
445 = (-19178)3 + 1504393 + 2*(-119321)3
445 = (-240572)3 + 3776053 - 2*2712563
By Kenji Koyama (February 21, 2000)
183 = 41700613 + (-4494438)3 + 2*(2090533)3
491 = 134766593 + 135849083 + 2*(-13531000)3
931 = (-6942368)3 + (-23115371)3 + 2*(18510833)3
(September 07, 2002)
Many new results for 1000 ≤ n ≤ 10000 were found by Jean-Charles Meyrignac.
He extended the search range up to 4,000,000.
(May 04, 2003)
Mike Oakes tried up to 4,000,000 again, and found 6 results.
3982 = -2 * 332669^3 - 11077811^3 + 11078011^3
5972 = 2 * 2980937^3 + 3002495^3 - 4309669^3
6124 = -2 * 209552^3 - 17513657^3 + 17513677^3
7151 = 2 * 3667541^3 + 3004190^3 - 5010331^3
8581 = -2 * 3842861^3 - 66333785^3 + 66342382^3
8653 = 2 * 30046^3 + 4252402^3 - 4252403^3
148, 671, 788 1084, 1121, 1247, 1444, 1462, 1588, 1975 2246, 2300, 2372, 2822 3047, 3268, 3307, 3335, 3380, 3641, 3676, 3956 4036, 4108, 4369, 4388, 4819, 4883, 4990 5188, 5279, 5468, 5540, 5620, 5629 6707, 6980 7097, 7106, 7132, 7177, 7323, 7519, 7708, 7727, 7799, 7853, 7862, 7988 8114, 8380, 8572, 8588, 8644, 8779, 8887, 8968 9274, 9463, 9589, 9724, 9850 |
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