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I came accross a problem in an excerpt of the book "1001 problems in classical number theory" that asks to determine all rational solutions to the equation :

$$ x^3 + y^3 = x^2 + y^2$$

Apart from the trivial solutions $(1,0)$ $(1,1)$, $(0,1)$ $(0,0)$, I couldn't find a way to determine all rational solutions.

Here are some other solutions found by some algorithm >> https://benvitalenum3ers.wordpress.com/2015/07/01/rational-number-solutions-to-x3-y3-x2-y2/

Any ideas are welcome.

Thanks

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I get that all solutions are the intersection of the curve (over the reals) with lines through $(1,1)$ with slope a negative square, that is $$ p^2 x + q^2 y = p^2 + q^2 $$ with intgers $p,q > 0$ and, might as well, $\gcd(p,q) = 1.$

ADDED: I went through it more carefully, for each positive pair we get two intersections. A little cleaner to allow integers $\gcd(p,q) = 1$ and $p + q \neq 0,$ not necessarily both positive. Then $$ x = \frac{ p^3 + pq^2\; }{p^3 + q^3}= \frac{ \left(p^2 + q^2\right)p \; }{p^3 + q^3} \; \; , $$ $$ y = \frac{p^2 q + q^3 \;}{p^3 + q^3} \; \; = \frac{ \left(p^2 + q^2\right)q \; }{p^3 + q^3} \; \; . $$

For that matter, it turns out that we can take any direction $(p,q)$ with integers $p,q$ and $p+q > 0$ ( so that we will also have $p^3 + q^3 > 0$) and simply adjust the length by multiplying by the correct scalar quantity, giving $$ (x,y) = \; \; \left( \frac{ \; p^2 + q^2 \; }{\; p^3 + q^3 \;} \right) \; \; \; (p,q) $$

Let me edit in the nine examples at LINK

For example, with $(p,q) = (3,-1),$ we get $\frac{ \; p^2 + q^2 \; }{\; p^3 + q^3 \;} = \frac{9+1}{27 - 1} = \frac{10}{26} = \frac{5}{13},$ so we get $(x,y) = \left(\frac{5}{13}\right) (3,-1) = \left(\frac{15}{13}, \; \; -\frac{5}{13} \right)$

$$ (3,-1) \mapsto \left( \frac{15}{13}, \; - \frac{5}{13} \right) $$ $$ (2,-1) \mapsto \left( \frac{10}{7}, \; - \frac{5}{7} \right) $$ $$ (5,-3) \mapsto \left( \frac{85}{49}, \; - \frac{51}{49} \right) $$ $$ (3,-2) \mapsto \left( \frac{39}{19}, \; - \frac{26}{19} \right) $$ $$ (7,-5) \mapsto \left( \frac{259}{109}, \; - \frac{185}{109} \right) $$ $$ (4,-3) \mapsto \left( \frac{100}{37}, \; - \frac{75}{37} \right) $$ $$ (9,-7) \mapsto \left( \frac{585}{193}, \; - \frac{455}{193} \right) $$ $$ (5,-4) \mapsto \left( \frac{205}{61}, \; - \frac{164}{61} \right) $$ $$ (11,-9) \mapsto \left( \frac{1111}{301}, \; - \frac{909}{301} \right) $$

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Indeed, all we need to do is take consecutive exponents, say $$ \color{red}{x^7 + y^7 = x^6 + y^6.} $$ We can take any direction $(u,v)$ with integers $u,v$ and $u+v > 0$ and multiply by the correct scalar quantity, giving $$ (x,y) = \; \; \left( \frac{ \; u^6 + v^6 \; }{\; u^7 + v^7 \;} \right) \; \; \; (u,v) $$

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The precise form of the numerator and denominator are not that important, as long as they are homogeneous. I would get an analogous outcome for $$\color{red}{ x^4 + x^2 y^2 + y^4 = x^3 + x^2 y + x y^2 + y^3}, $$ all rational points are $$ (x,y) = \; \; \left( \frac{ \; u^3 + u^2 v + u v^2 + v^3 \; }{\; u^4 + u^2 v^2 + v^4 \;} \right) \; \; \; (u,v) $$ enter image description here

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  • $\begingroup$ Let's have an example? $\endgroup$ – AmateurMathPirate Sep 1 '17 at 22:08
  • $\begingroup$ @AmateurMathGuy example of what? $\endgroup$ – Will Jagy Sep 1 '17 at 22:17
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    $\begingroup$ A non trivial example of an $(x,y) $ that answers the question $\endgroup$ – AmateurMathPirate Sep 1 '17 at 22:20
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    $\begingroup$ @AmateurMathGuy For example, with $(p,q) = (3,-1),$ we get $\frac{ \; p^2 + q^2 \; }{\; p^3 + q^3 \;} = \frac{9+1}{27 - 1} = \frac{10}{26} = \frac{5}{13},$ so we get $(x,y) = \left(\frac{5}{13}\right) (3,-1) = \left(\frac{15}{13}, \; \; -\frac{5}{13} \right)$ $\endgroup$ – Will Jagy Sep 1 '17 at 22:29

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