Diophantine: $x^3+y^3=z^3 \pm 1$
There are infinitely many solutions.
Parametric subset of solutions for $x^3+y^3=z^3+1$:
$x=9n^3+1$,
$y=9n^4$, $\qquad\qquad\qquad$ $(n\in \mathbb{N})$;
$z=9n^4+3n$.
Parametric subset of solutions for $x^3+y^3=z^3-1$:
$x=9n^3-1$,
$y=9n^4-3n$, $\qquad\qquad\qquad$ $(n\in \mathbb{N})$;
$z=9n^4$.
Of course, there are other solutions too (out of described subsets):
$64^3+94^3=103^3+1$, $\qquad$ $135^3+235^3=249^3+1$, $\quad\ldots\;$;
$135^3+138^3=172^3-1$, $\quad$ $372^3+426^3=505^3-1$, $\quad\ldots\;$.
Explanation:
How to obtain these subsets?
I printed starting solutions (using brute force) of equation
$x^3+y^3=z^3+1$.
$(x,y,z) : \alpha$, $\qquad$ where $x\leqslant y, \quad \alpha = y/x$;
$(9, 10, 12) : 1.11111$
$(64, 94, 103) : 1.46875$
$(73, 144, 150) : \color{#FF2200}{1.9726}$
$(135, 235, 249) : 1.74074$
$(334, 438, 495) : 1.31138$
$(244, 729, 738) : \color{#FF2200}{2.9877}$
$(368, 1537, 1544) : 4.17663$
$(1033, 1738, 1852) : 1.68248$
$(1010, 1897, 1988) : 1.87822$
$(577, 2304, 2316) : \color{#FF2200}{3.99307}$
$(3097, 3518, 4184) : 1.13594$
$(3753, 4528, 5262) : 1.2065$
$(1126, 5625, 5640) : \color{#FF2200}{4.99556}$
$(4083, 8343, 8657) : 2.04335$
$(5856, 9036, 9791) : 1.54303$
$(3987, 9735, 9953) : 2.44169$
$(11161, 11468, 14258) : 1.02751$
$(1945, 11664, 11682) : \color{#FF2200}{5.99692}$
$(13294, 19386, 21279) : 1.45825$
$(3088, 21609, 21630) : \color{#FF2200}{6.99773}$
$(10876, 31180, 31615) : 2.86686$
$(27238, 33412, 38599) : 1.22667$
$(27784, 35385, 40362) : 1.27357$
$(16617, 35442, 36620) : 2.13288$
$(4609, 36864, 36888) : \color{#FF2200}{7.99826}$
$\ldots$
and observed that some $\alpha$ are almost integer (I denoted them by red color).
So, I created "red list" $-$ subset of solutions:
$\color{#AAAAAA}{(10, 9, 12) : 0.9}$
$(73, 144, 150) : \color{#FF2200}{1.9726}$
$(244, 729, 738) : \color{#FF2200}{2.9877}$
$(577, 2304, 2316) : \color{#FF2200}{3.99307}$
$(1126, 5625, 5640) : \color{#FF2200}{4.99556}$
$(1945, 11664, 11682) : \color{#FF2200}{5.99692}$
$(3088, 21609, 21630) : \color{#FF2200}{6.99773}$
$(4609, 36864, 36888) : \color{#FF2200}{7.99826}$
$\ldots$
Then observed regularity $(9n^3+1,9n^4,9n^4+3n)$.
It is easy to prove that it is true:
$z^3+1-x^3-y^3 = (9n^4+3n)^3 + 1 - (9n^3+1)^3 - (9n^4)^3 =$
$(729n^{12}+729n^9+243n^6+27n^2) - 1 - (729n^9+243n^6+27n^3+1) - 729n^{12} = 0.
$
Same way $-$ for equation $x^3+y^3=z^3-1$.
Note that Noam Elkies has given a complete parametrization of the integer solutions of $x^{3} + y^{3} +z^{3}+w^{3}=0,$ which may be found at http://www.math.harvard.edu/~elkies/4cubes.html .
I think we've had this one before. The one I remember is $10^3 + 9^3 = 12^3 + 1.$ I would expect there to be polynomial families of solutions $x=f(t), y=g(t), z=h(t).$ Other than that, note that for any fixed $z,$ the set of $(x,y)$ pairs is finite because $$ | x^3 + y^3 | = |x+y| (x^2 - xy+ y^2), $$ where the quadratic form $(x^2 - xy+ y^2)$ is positive definite, indeed $$ x^2 - xy+ y^2 \geq 3 x^2 / 4, $$ $$ x^2 - xy+ y^2 \geq 3 y^2 / 4. $$ I would not expect to be able to write all solutions by formula.
You ought to run a computer search, say $0 \leq |y| \leq x < M,$ see when $x^3 + y^3 + 1$ or $x^3 + y^3 - 1$ are cubes, some large $M$ depending on your computer and language.
A different search, still exhaustive, is just to let $z \geq 2$ increase, find both $z^3 -1$ and $z^3 + 1,$ for each find all $(x,y)$ pairs solving your equation, and print out those when $x \neq z$ and $y \neq z.$ You can build in any restriction that might speed it up, factoring and so on. This seems worth doing, I think I'll try it for small numbers.
=================================
z x y x/y
9 -1 8 6 1.333333333333333
12 1 10 9 1.111111111111111
103 1 94 64 1.46875
144 -1 138 71 1.943661971830986
150 1 144 73 1.972602739726027
172 -1 138 135 1.022222222222222
249 1 235 135 1.740740740740741
495 1 438 334 1.311377245508982
505 -1 426 372 1.145161290322581
577 -1 486 426 1.140845070422535
729 -1 720 242 2.975206611570248
738 1 729 244 2.987704918032787
904 -1 823 566 1.454063604240283
1010 -1 812 791 1.026548672566372
1210 -1 1207 236 5.114406779661017
1544 1 1537 368 4.176630434782608
1852 1 1738 1033 1.682478218780252
1988 1 1897 1010 1.878217821782178
2304 -1 2292 575 3.986086956521739
2316 1 2304 577 3.993067590987868
3097 -1 2820 1938 1.455108359133127
3753 -1 3230 2676 1.207025411061285
4184 1 3518 3097 1.135938004520504
5262 1 4528 3753 1.206501465494271
5625 -1 5610 1124 4.991103202846976
5640 1 5625 1126 4.995559502664299
6081 -1 5984 2196 2.724954462659381
6756 -1 6702 1943 3.449305198147195
8657 1 8343 4083 2.043350477590007
8703 -1 8675 1851 4.68665586169638
9791 1 9036 5856 1.543032786885246
9953 1 9735 3987 2.441685477802859
11664 -1 11646 1943 5.993823983530623
11682 1 11664 1945 5.996915167095116
12884 -1 11903 7676 1.550677436164669
14258 1 11468 11161 1.027506495833707
16849 -1 16806 3318 5.065099457504521
18649 -1 17328 10866 1.594699061292104
21279 1 19386 13294 1.458251842936663
21609 -1 21588 3086 6.99546338302009
21630 1 21609 3088 6.997733160621761
24987 -1 24965 3453 7.229944975383725
29737 -1 27630 17328 1.594529085872576
31615 1 31180 10876 2.866862817212211
36620 1 35442 16617 2.132875970391768
36864 -1 36840 4607 7.99652702409377
36888 1 36864 4609 7.998264265567368
37513 -1 31212 28182 1.107515435384288
38134 -1 37887 10230 3.703519061583578
38239 -1 33857 25765 1.314069474092761
38599 1 33412 27238 1.226668624715471
38823 1 38782 5700 6.803859649122807
40362 1 35385 27784 1.273574719262885
41485 1 41167 11767 3.498512790005949
41545 -1 34566 31212 1.107458669742407
47584 1 44521 26914 1.654194842832726
49461 -1 49409 7251 6.814094607640325
51762 -1 46212 34199 1.351267580923419
57978 1 51762 38305 1.351311839185485
59049 -1 59022 6560 8.997256097560976
59076 1 59049 6562 8.998628466930814
63086 1 50920 49193 1.035106620860692
66465 -1 66198 15218 4.349980286502825
68010 -1 66167 29196 2.266303603233319
69709 -1 56503 54101 1.044398439954899
71852 -1 69479 32882 2.112979745757557
73627 -1 64165 51293 1.250950422084885
73967 1 72629 27835 2.60926890605353
78529 -1 78244 17384 4.500920386562356
79273 1 76903 35131 2.189035324926703
83711 1 83692 7364 11.36501901140684
83802 1 67402 65601 1.027453849788875
86166 1 80020 50313 1.590443821676306
90000 -1 89970 8999 9.99777753083676
90030 1 90000 9001 9.998889012331963
95356 -1 87383 58462 1.494697410283603
108608 -1 94904 75263 1.260964883143112
109747 -1 89559 84507 1.059782029891015
128692 -1 104383 99800 1.045921843687375
131769 -1 131736 11978 10.9981632993822
131802 1 131769 11980 10.99908180300501
Mon Jun 10 15:07:10 PDT 2013
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