Statement

For some positive integers k, there exists an integer partition of the form 4^t = 2^t + k,
where 4^t, 2^t, and k are all positive integers and t is a real number.

The first two such partitions are 4¹ = 2¹ + 2 and 4^1.5849625… = 2^1.5849625… + 6.

Partitions where t is also an integer are called perfect.
For any m 1 let P(m) be the proportion of such partitions that are perfect with p $\leq$ m.
Thus P(6) = 1/2.

In the following table are listed some values of P(m)

P(5) = 1/1
P(10) = 1/2
P(15) = 2/3
P(20) = 1/2
P(25) = 1/2
P(30) = 2/5
…
P(180) = 1/4
P(185) = 3/13

Find the smallest m for which P(m) 1/12345

Solution

Let's start to process the equation:

It is easy to notice that for whatever 2^t(which has to be integer) there will exist a corresponding k that will be integer. And like $4^{t} = {2^{t}}^{2}}$ then 4^t will be integer. As t can be real we will have a valid solution to the equation for every t that 2^t is integer. And the solution is perfect when t is integer too.

It is also important to notice that the function that defines k is increasing so for every t₁ < t₂ => k₁ < k₂.

Knowing that, it is easy to program an increasing function that counts the proportion of perfect partitions until the value we are looking for is reached.

from fractions import Fraction
from math import log
 
calc_k = lambda x: x * (x - 1)
 
cant_perf = 1
cant_tot = 1
limit = Fraction(1, 12345)
 
if __name__ == '__main__':
    i = 3
    while Fraction(cant_perf, cant_tot) >= limit:
        log_2 = int(log(i,2))
        if 2 ** log_2 == i:
            cant_perf += 1
        cant_tot += 1
        i += 1
    i -= 1
    print("The result is:", calc_k(i))

The Python file is available for download here.