System of equations : x^n + y^n, n = 1,2,3

 
 
x, y   are positive integers such that

x \; + \; y \; = \; a^2
x^2 \; + \; y^2 \; = \; b^2
x^3 \; + \; y^3 \; = \; c^2

 

For example,

(1)

184 \; + \; 345 \; = \; 23^2
184^2 \; + \; 345^2 \; = \; 391^2
184^3 \; + \; 345^3 \; = \; 6877^2

and,   184^2 \; + \; 345^2 \; - \; (184\times 345) \; = \; 299^2
 

(2)

736 \; + \; 1380 \; = \; 46^2
736^2 \; + \; 1380^2 \; = \; 1564^2
736^3 \; + \; 1380^3 \; = \; 55016^2

and,   736^2 \; + \; 1380^2 \; - \; (736\times 1380) \; = \; 1196^2

x^3 \; + \; y^3 \; = \; (x+y) \,(x^2-x \, y+y^2)

If   x^3+y^3   is a square and   (x+y)   is a square, then so is   (x^2-x \, y+y^2)

If we write   m = x(x + y)   and   n = y(x + y)

m \; + \; n \; = \; x(x + y) + y(x + y) \; = \; (x + y)^2

m^2 \; + \; n^2 \; = \; (x(x + y))^2 + (y(x + y))^2 \; = \; (x + y)^2 \, (x^2 + y^2)

m^3 \; + \; n^3 \; = \; (x(x + y))^3 + (y(x + y))^3
    = \; (x+y)^3 \, (x^3+y^3) \; = \; (x+y)^4 \, (x^2-x \, y+y^2)

So, if we can find   x   and   y   such that   x^2+y^2   and   x^2-x \, y+y^2   are squares, we will be able to construct parametric solutions to the system of equations.

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

3 Responses to System of equations : x^n + y^n, n = 1,2,3

  1. paul says:

    Here are a few with x<=50000

    184 + 345 = 23^2
    184^2 + 345^2 = 391^2
    184^3 + 345^3 = 6877^2

    736 + 1380 = 46^2
    736^2 + 1380^2 = 1564^2
    736^3 + 1380^3 = 55016^2

    1656 + 3105 = 69^2
    1656^2 + 3105^2 = 3519^2
    1656^3 + 3105^3 = 185679^2

    2944 + 5520 = 92^2
    2944^2 + 5520^2 = 6256^2
    2944^3 + 5520^3 = 440128^2

    4600 + 8625 = 115^2
    4600^2 + 8625^2 = 9775^2
    4600^3 + 8625^3 = 859625^2

    6624 + 12420 = 138^2
    6624^2 + 12420^2 = 14076^2
    6624^3 + 12420^3 = 1485432^2

    9016 + 16905 = 161^2
    9016^2 + 16905^2 = 19159^2
    9016^3 + 16905^3 = 2358811^2

    11776 + 22080 = 184^2
    11776^2 + 22080^2 = 25024^2
    11776^3 + 22080^3 = 3521024^2

    14904 + 27945 = 207^2
    14904^2 + 27945^2 = 31671^2
    14904^3 + 27945^3 = 5013333^2

    18400 + 34500 = 230^2
    18400^2 + 34500^2 = 39100^2
    18400^3 + 34500^3 = 6877000^2

    22264 + 41745 = 253^2
    22264^2 + 41745^2 = 47311^2
    22264^3 + 41745^3 = 9153287^2

    26496 + 49680 = 276^2
    26496^2 + 49680^2 = 56304^2
    26496^3 + 49680^3 = 11883456^2

    31096 + 58305 = 299^2
    31096^2 + 58305^2 = 66079^2
    31096^3 + 58305^3 = 15108769^2

    36064 + 67620 = 322^2
    36064^2 + 67620^2 = 76636^2
    36064^3 + 67620^3 = 18870488^2

    41400 + 77625 = 345^2
    41400^2 + 77625^2 = 87975^2
    41400^3 + 77625^3 = 23209875^2

    47104 + 88320 = 368^2
    47104^2 + 88320^2 = 100096^2
    47104^3 + 88320^3 = 28168192^2

    Paul.

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