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 ?

Solution 1:

According to SAGE:

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.
$P$ 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:

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

If $a+b+c=3$, then $ \frac{a}{5b+c^3}+\frac{b}{5c+a^3}+\frac{c}{5a+b^3} \geq \frac{1}{2}$

Linear independence of the numbers $\{1,\pi,{\pi}^2\}$

How prove $a_{n}=1!+2!+\cdots+n!$ contains infinitely many prime factors

Show that convolution of two measurable functions is well-defined

Does $n \mid 2^{2^n+1}+1$ imply $n \mid 2^{2^{2^n+1}+1}+1$?

$x,y,z$ positive real numbers , $x+y+z=3$ $\implies x^4y^4z^4(x^3+y^3+z^3)≤3$

linearity of expectation in case of dependent events

Is $\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$ a rational number for $m,n\ge 2\in\mathbb N$?

Rigorous separation of variables.

Is the value of $\sin(\frac{\pi}{n})$ expressible by radicals?

How many Groups there are on a finite set?

$f$ is a non-constant polynomial, $A $ is a set of measure zero, Is this true that $m(f^{-1}A)=0$, where $m$ stands for the Lebesgue measure.