## Equation : x^3 + y^3 = z^3 ± 1

$x^3 \; + \; y^3 \; = \; z^3 \; \pm \; 1$

where   $x, \; y, \; z$   are integers

For example,

$4528^3 \; + \; 3753^3 \; = \; 5262^3 \; + \; 1$

$2676^3 \; + \; 3230^3 \; = \; 3753^3 \; - \; 1$

Find smaller values for   x,   y,   and   z.

We also have solutions of the form

$A^3 \; + \; B^3 \; = \; C^3 \; + \; D^3 \; = \; E^3 \; - \; 1$

$(-4528)^3 + 5262^3 = 2676^3 + 3230^3 = 3753^3 - 1$

$(-837313192)^3 + 972979926^3 = 597125510^3 + 494833692^3 = 693875529^3 - 1$

$(-154817534664928)^3 \; + \; 179902042398942^3$
$= \; 110407312449710^3 \; + \; 91493760027396^3$
$= \; 128296197510633^3 \; - \; 1$

$33263527834506644646^3 \; + \; (-28625452523638639192)^3$
$= \; 16917013241050678572^3 \; + \; 20414091256729259030^3$
$= \; 23721710326627098249^3 \; - \; 1$

$(-5292788920560919574752528)^3 \; + \; 6150359769364707539403822^3$
$= \; 3774524645076319223584190^3 \; + \; 3127921914152294606624916^3$
$= \; 4386096795844401014486313^3 \; - \; 1$

math grad - Interest: Number theory
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### 2 Responses to Equation : x^3 + y^3 = z^3 ± 1

1. paul says:

Here are the smaller one

10^3 + 9^3 =12^3 + 1
94^3 + 64^3 =103^3 + 1
144^3 + 73^3 =150^3 + 1
235^3 + 135^3 =249^3 + 1
438^3 + 334^3 =495^3 + 1
729^3 + 244^3 =738^3 + 1
1537^3 + 368^3 =1544^3 + 1
1738^3 + 1033^3 =1852^3 + 1
1897^3 + 1010^3 =1988^3 + 1
2304^3 + 577^3 =2316^3 + 1

and

8^3 + 6^3 = 9^3 – 1
138^3 + 71^3 = 144^3 – 1
138^3 + 135^3 = 172^3 – 1
426^3 + 372^3 = 505^3 – 1
486^3 + 426^3 = 577^3 – 1
720^3 + 242^3 = 729^3 – 1
812^3 + 791^3 = 1010^3 – 1
823^3 + 566^3 = 904^3 – 1
1207^3 + 236^3 = 1210^3 – 1
2292^3 + 575^3 = 2304^3 – 1

Paul.

• benvitalis says:

Nice. Thanks for supplying answers. Note that we have solutions of the form
$A^3 \; + \; B^3 \; = \; C^3 \; + \; D^3 \; = \; E^3 \; - \; 1$