How to determine if the graphs, ${2}^p$ and q! are close enough

Question

There are two graphs ${2}^p$ and q! and we need to determine if they are close enough at some point. Mathematically, it was something like this

1< ${2}^p$ /q! <1.1

So in other terms, is it possible to check if there are any integral values of p(p>7, p $\in$ ${I}^+$) and q(q>5, q $\in$ ${I}^+$) due to which the division gives a value between 1 and 1.1

I have tried figuring it out using the graphical approach and I think there doesn't exists any p or q such that p>7 and q>5, but I think there exists a concrete mathematical approach to figure it out.

Answer 1

Stirling's approximation is enough to generate solutions.

$$2^x \approx y! \approx \sqrt{2\pi y}\left(\frac ye \right)^y \implies x \approx \frac12 \log_2 (2\pi y) + y \log_2 \left(\frac ye \right)$$

These are all pairs $(x,y)$ which satisfy your inequality with $y \le 100$:

{{7, 5}, {70, 22}, {98, 28}, {133, 35}, {170, 42}, {203, 48}, {220, 51}, {290, 63}, {296, 64}, {351, 73}, {376, 77}, {414, 83}, {459, 90}, {505, 97}}

For each of these pairs,

$$ \left\lfloor \frac12 \log_2 (2\pi y) + y \log_2 \left(\frac ye \right) \right\rfloor \le x \le \left\lceil \frac12 \log_2 (2\pi y) + y \log_2 \left(\frac ye \right) \right\rceil. $$

Using this bound you can generate pairs as large as you want through some trial/error. For instance, here are some pairs around $y = 10^6$:

{18489124, 1000012}, {18489144, 1000013}

Edit: In hindsight, a built-in loggamma function might be faster than Stirling's.

Answer 2

It is trivial to solve by computer. Here are the first few results (where $r$ is the ratio $2^p/q!$):

  p=1  q=2  r=1
  p=7  q=5  r=1.06666666666667
  p=70  q=22  r=1.05034773691979
  p=98  q=28  r=1.03943839001342
  p=133  q=35  r=1.05379655617941
  p=170  q=42  r=1.06517520295244
  p=203  q=48  r=1.03557207701665
  p=220  q=51  r=1.08631055044999

The program simply alternates between incrementing $q$ (making $q!>2^p$), and incrementing $p$ one or more times (until $2^p>q!$), and then checking whether the ratio is small enough. It is of course better to take logs so as to not work with such large numbers. The ratio can get quite small, for example

  p=4283  q=557  r=1.00016494818775