## 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.

math grad - Interest: Number theory
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### 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.

• benvitalis says:

Basically, we need to find (x,y) such that x^2 + y^2 and x^2 – x*y + y^2 are both squares.
Can you write parametric solutions?