Find all integer solutions of $a^3 + b^2 = 100000$ ?

I'm looking for one solution and get idea from that to write an analytic solution, but I've not found yet. Is it a good idea or I should start it analytically. If so how to start ?

  • 1
    $\begingroup$ What I noticed is that the last digits of $a$ and $b$ should be one of these : $(0,0), (1,3), (1,7), (4,4), (4,6), (5,5), (6,2), (6,8), (9,1), (9,9)$. If this is useful I dont know. $\endgroup$ – Integral Jul 25 '13 at 6:09
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    $\begingroup$ @Mahdi Khosravi: see this article about your question math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf $\endgroup$ – M.H Jul 25 '13 at 6:10
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    $\begingroup$ One solution is $a=-41$, $b=\pm 411$, which is the only integral point found by Magma. $\endgroup$ – Erick Wong Jul 25 '13 at 6:49
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    $\begingroup$ I have written a computer algorithm which computes the solutions via a modulo attack. You could also brute force. I think the only solutions are $$a: -70833, b:+5561,-5561$$ and $$a:-41 b:+411,-411$$ $\endgroup$ – Torsten Hĕrculĕ Cärlemän Jul 25 '13 at 16:52
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    $\begingroup$ @TorstenHĕrculĕCärlemän I think you have integer overflow. The first pair works mod $2^{32}$, but clearly $a=-70833$ is far too large to match with $b=5561$. $\endgroup$ – Erick Wong Jul 25 '13 at 18:12

According to SAGE:

  1. The elliptic curve $E: y^2=x^3+100000$ has rank $1$ over $\mathbf Q$. It is generated by the integral point $P=(41, 411)$, found by Erick in the comments.
  2. $P$ is the only integral point on $E$.

(Remark: these are not approximate results of the form "$P$ is the only integral point SAGE could find before my motherboard exploded". They are actual end-of-the-question results.)

Other rational points on $E$ include:

$$2P = \left(-\frac{3330471}{75076}, \frac{2318226083}{20570824}\right)$$ $$3P = \left(\frac{639610632355481}{41069987336569} ,-\frac{84788682808343092092621}{263200586935300718003}\right).$$

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