I am trying to find smallest number s.t. it's square has more than 8 million divisors. It might seem impossible for hand and even while solving with a computer, it requires great amount of time (but I have been able to establish that the number is greater than 2147483647 till now). I have also plotted some graphs and I think there is some pattern in the numbers which have large number of divisors nad it seems like they occur on periodic distances: (Note: $\sigma_x(n)$ is Divisor Function)
$$y=\sigma_0(x^2)$$ What could be a possible approach to finding that number, if required using a computer too.
We notice that we need to find: $$N=\prod^n p_i^{q_i}$$ where: $$\prod^n(2q_i+1)>8\times10^6$$ while minimizing N.
It is easy to exhibit such a number, for example $2^{8000000}$ is the square of $2^{4000000}$. But perhaps you want the smallest number in which case the number of divisors of $N=\prod p_i^{a_i}$ where the $p_i$ are distinct primes is $\prod (a_i+1)$. You need all the $a_i$ to be even. If you choose $a_i=2$ note that $3^{15}\gt 8000000$ so you will need fifteen primes. Or $3^{6}\times 5^6\gt 8000000$ with twelve primes.
So using this analysis you can search for the least square with more than $8000000$ divisors. It is easy to make this a finite search.