Find the sum of all positive integers m such that 2^m can be expressed as sums of four factorials (of positive integers).
To solve this problem, we can start by listing out the first few powers of 2 and the factorials of positive integers:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
...
n! = 1, 2, 6, 24, ...
Now, we need to find the positive integers m such that 2^m can be expressed as the sum of four factorials. Let's try to find some patterns.
For 2^1 = 2, there is no combination of four factorials that adds up to 2.
For 2^2 = 4, there is still no combination of four factorials that adds up to 4.
For 2^3 = 8, we can find a combination: 8 = 3! + 2! + 1! + 1!.
For 2^4 = 16, there is no combination of four factorials that adds up to 16.
We can observe that as the powers of 2 increase, it becomes more difficult to find combinations of four factorials that add up to those values.
To approach this problem systematically, we can start checking the possible values of m by increasing it one by one and checking if there is a combination of four factorials that adds up to 2^m. We can stop when we find no more combinations.
m = 3: We found the combination 2^3 = 8 = 3! + 2! + 1! + 1!.
m = 4: There is no combination of four factorials that adds up to 2^4 = 16.
Therefore, the only positive integer m that satisfies the condition is m = 3.
The sum of all positive integers m that satisfy the condition is 3.