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

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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.

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