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

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 ?

• 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. – Integral Jul 25 '13 at 6:09
• One solution is $a=-41$, $b=\pm 411$, which is the only integral point found by Magma. – Erick Wong Jul 25 '13 at 6:49
• 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 $$a:-41 b:+411,-411$$ – Torsten Hĕrculĕ Cärlemän Jul 25 '13 at 16:52
• @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 $b=5561$. – Erick Wong Jul 25 '13 at 18:12

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