Find the sum of all positive integers m such that 2^m can be expressed as sums of four factorials (of positive integers).
Details and assumptions
The number n!, read as n factorial, is equal to the product of all positive integers less than or equal to n. For example, 7!=7×6×5×4×3×2×1.
The factorials do not have to be distinct. For example, 2^4=16 counts, because it equals 3!+3!+2!+2!.
To find the sum of all positive integers m such that 2^m can be expressed as sums of four factorials, we'll go through the possible values of m step-by-step.
Step 1: Observe the pattern
Let's look at the powers of 2 and see if we notice any patterns:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
...
We can see that every power of 2 is a doubling of the previous power. This is important to keep in mind for the next steps.
Step 2: Understand how factorials work
In order to express a power of 2 as a sum of four factorials, it means that we need to find four factorials that add up to that power of 2.
For example, let's take 2^4 = 16. To represent this as a sum of four factorials, we can choose any combination of factorials that add up to 16. One possible combination is 3! + 3! + 2! + 2! = 6 + 6 + 2 + 2 = 16.
Step 3: Analyze the factorials involved
To better understand the factorials involved, let's consider the maximum value of each factorial that can be used (the largest positive integer for each factorial).
- 3! = 6 (since 3! = 3 × 2 × 1 = 6)
- 2! = 2 (since 2! = 2 × 1 = 2)
- 1! = 1 (since 1! = 1 × 0! = 1 × 1 = 1)
- 0! = 1 (by definition)
We can see that the maximum value for each factorial is less than or equal to the next factorial. This tells us that we need to consider the maximum factorials first, starting with 3!.
Step 4: Identify possible combinations
We need four factorials to sum up to each power of 2. Let's go through the possible combinations of these factorials for each power of 2:
For 2^1 = 2:
- There are no combinations of factorials that sum up to 2.
For 2^2 = 4:
- Only one possible combination: 2! + 2! + 0! + 0! = 2 + 2 + 1 + 1 = 6
For 2^3 = 8:
- Only one possible combination: 3! + 0! + 0! + 0! = 6 + 1 + 1 + 1 = 9
For 2^4 = 16:
- Multiple possible combinations exist:
- 3! + 3! + 2! + 2! = 6 + 6 + 2 + 2 = 16
- 4! + 1! + 1! + 1! = 24 + 1 + 1 + 1 = 27
- 4! + 0! + 0! + 0! = 24 + 1 + 1 + 1 = 27
For higher powers of 2, it follows the same pattern as above.
Step 5: Calculate the sum of possible values of m
Now, let's calculate the sum of the values of m that correspond to the possible combinations we found in Step 4:
For 2^1 = 2:
- No combinations, so m = 0
For 2^2 = 4:
- Only one combination: m = 2
For 2^3 = 8:
- Only one combination: m = 3
For 2^4 = 16:
- Multiple combinations: m = 4, m = 4 (same value)
The total sum is m = 0 + 2 + 3 + 4 + 4 = 13.
Therefore, the sum of all positive integers m such that 2^m can be expressed as a sum of four factorials is 13.
To find the sum of all positive integers m that satisfy the given condition, we can use a combination of brute force and systematic analysis.
First, let's observe that the largest factorial term we need to consider is 3!. This is because any factorial greater than 3! can be expressed as a sum of 2!, 3!, and less than 3! using the increasing nature of factorial values.
Now, let's generate a list of powers of 2 up to a reasonable limit. We can start from 2^1 and keep doubling until we reach a value that is greater than our largest possible sum of factorials.
Starting with 2^1, we have:
2^1 = 2
Next, we have:
2^2 = 4
Next, we have:
2^3 = 8
Next, we have:
2^4 = 16 <-- This is the first power of 2 greater than 3!+3!+2!+2!
We can stop here because any larger power of 2 will exceed our largest possible sum of factorials.
Now, let's analyze the powers of 2 we have obtained so far. We need to check if each power of 2 can be expressed as the sum of four factorials.
For 2^1 = 2, there is no way to express it as the sum of four positive factorials.
For 2^2 = 4, there is no way to express it as the sum of four positive factorials.
For 2^3 = 8, there is no way to express it as the sum of four positive factorials.
For 2^4 = 16, we can express it as 3! + 3! + 2! + 2!.
Therefore, the only positive integer m that satisfies the condition is m = 4. Thus, the sum of all positive integers m is 4.
In summary, we start from 2^1 and generate powers of 2 until we reach a value larger than the largest possible sum of factorials. Then, we check if each power of 2 can be expressed as the sum of four factorials. Finally, we sum up the values of m that satisfy the condition.