<s>[INST] The solutions of the equation $|p^3-q^2|=2^n$ Let's consider the following equation
$$|p^3-q^2|=2^n$$
where $p$, $q$ are distinct primes and $n$ a non-negative integer. The equation admits some trivial solutions:
$$|2^3-3^2|=2^0$$
$$|3^3-5^2|=2^1$$
$$|5^3-11^2|=2^2$$
$$|17^3-71^2|=2^7$$
Clearly, not all values of $n$ admit a solution (for example, if $3|n$ the equation has no solution).
Are there other solutions besides those mentioned above?
Many thanks.
 [/INST] We consider the elliptic curves  $y^2 = x^3\pm2^n$ with parameter $n$ where $x$ and $y$ are prime number .
Searching the integer points $(x,y)$ with $\boldsymbol{n \lt 100}$ and $\boldsymbol{x \lt 10^{10}}$, we found the integer points $(x,y,n)=(3, 5, 1)$ and $(5, 11, 2)$ for $y^2 = x^3-2^n.$
Similarly, we found the integer point $(x,y,n)= (17, 71, 7))$ for $y^2 = x^3+2^n.$
There seems to be no other solution.
</s>