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

Question

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 ?

Answer 1

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:

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