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How many nontrivial integer solutions does $x^3+y^3=z^3 \pm 1$ have?

The trivial solutions are $(\pm 1,z,z)$ and $(z,\pm 1,z)$.

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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 $x^3+y^3=z^3-1$.

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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 .

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  • $\begingroup$ I wish I understood what he was talking about ($P^2$ blown up? Blown down lines?). My background in Number Theory is very elementary to say the least. But still, thanks for the link. $\endgroup$ – genepeer Jun 9 '13 at 21:03
  • $\begingroup$ @genepeer: I just saw your comment. I don't think you need to understand all that to understand the statement of what the solutions are. $\endgroup$ – Geoff Robinson Oct 31 '15 at 0:45
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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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