The parent question that maybe still remains to be answered at this moment is:
Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$ .
As far as the parent question is concerned, when generalizing to integer values $\;-\infty < x < +\infty$ , it is found that $(0,0)$ and indeed $(10,4)$ are integer solutions ( i.e. the only ones that I have been able to find, with some simple numerical means ).
If we solve $b$ from this equation, rename $b$ to $y$ and $a$ to $x$ , then the following real-valued function results: $$y = \ln\left[x^2\left(2^x-x^3\right)+1\right]/\ln(7) $$ Graphs of the function are depicted below. Red dots indicate integer values $(x,y)$ . The argument of the logarithm is colored "silver". There are two large red dots $(M,S)$ at the known integer solutions $(0,0)$ and $(10,4)$ of the Diophantine equation.
Viewport sizes from left to right / top to bottom:
xmin := -200; xmax := 200; ymin := -200; ymax := 200;
xmin := -10; xmax := 20; ymin := -15; ymax := 15;
xmin := -2; xmax := 2; ymin := -2; ymax := 2;
xmin := 9.90; xmax := 11; ymin := -0.1; ymax := 6;
- There are three zeroes $(M,P,Q)$ ; $M = (0,0)$ is easy. But were are $P$ and $Q$ ?
- There are two extreme values $(M,N)$ ; $M = (0,0)$ is easy. But what is $N=(x_N,y_N)$ ?
- There are two vertical asymptotes $(A,B)$. Between $A$ and $B$ the function is undefined.
Where are $x_A$ and $x_B$ ?
- The steep slope on the right changes into a moderate one. A straight line it seems. Is that right? If so, what then is the slope of that straight line?