For natural numbers $n$ and $k$, where $n \ge k$, let $n\underset{k \text{ times}}{\underbrace{!!!\dots!}}=n(n-k)(n-2k)\dots $ , where all the factors are positive integers (excluding zero and negative factors).
What is the greatest integer $n$ for which $n\underset{17 \text{ times}}{\underbrace{!!!\dots!}}<1025\underset{65 \text{ times}}{\underbrace{!!!\dots!}}$
The only idea I got is trial and error. To start by evaluating the right side of the inequality, which is approximately $9.27999 \times 10^{41}$. Then check for each of the following: $n=$RHS, $n(n-17)=$RHS, $n(n-17)(n-34)=$RHS, ... and so on, where $n=17,18,19,\dots$.
But this idea, in my opinion, is not feasible since we will have equations with orders higher than $4$ which have no solution in radicals to general polynomial equations with arbitrary coefficients (Abel–Ruffini theorem).
Can we solve this problem by the help of scientific calculators such as (CASIO fx-991es) but not by computers or websites?
I do not know how to start solving this problem. I am not asking you to solve it, I am asking for (help, hint, useful formulae) to be able to start. Thanks in advance!