Answer:
If repeats are allowed: [tex]5[/tex]
If no repeats are allowed: [tex]13[/tex]
Step-by-step explanation:
Natural numbers are a set of positive integers (1, 2, 3, etc.). Call [tex]s[/tex] a set of four natural numbers [tex]\displaystyle s=\{s_1, s_2, s_3, s_4\}[/tex] such that [tex]s_i\in \mathbb{N}[/tex].
Excluding 2, prime numbers must be odd. The only way to achieve a sum that is odd is if there is an odd number of odd numbers. Therefore, 2 must be one of the natural numbers, because it's not possible to obtain an odd sum with four odd numbers:
[tex]s=\{2, s_2, s_3, s_4\}[/tex]
The minimum sum of all four numbers requires minimizing the value of as many natural numbers as we can.
Recall the list of prime numbers (2, 3, 5, 7, 11, 13, etc.)
Since we know 2 must be part of the set, we can simply plug in numbers to try and get a sum of any prime number with two, three, and four numbers of the set.
[If repeats are allowed]
Since 1 is the smallest natural number, let the other three numbers be 1. Assuming repeats allowed, we have:
[tex]\text{Let } s=\{1, 1, 1, 2\}:\\ 1+1=2=\text{prime}\checkmark\\1+1+1=3=\text{prime}\checkmark\\ 2+1+1+1=5=\text{prime}\checkmark[/tex]
Since we've created a prime sum with two, three, and four numbers, we can see that the set [tex]s=\{1, 1, 1, 2\}[/tex] works. The sum of the four numbers in this set is [tex]\boxed{5}[/tex].
[If repeats are NOT allowed]
From the explanation above, 2 must still be one of the numbers in the set. Since we want to minimize each value, the smallest three natural numbers are 1, 2, 3. Let [tex]s=\{1, 2, 3, s_4\}[/tex] and solve for [tex]s_4[/tex] using your knowledge that:
- [tex]s_4[/tex] must be odd (you need an odd number of odd numbers to create a sum that is odd, so we need 1 or 3 odds to ensure that the sum of all four are odd and therefore prime)
- [tex]s_4\in \mathbb{N}[/tex] ([tex]s_4[/tex] needs to be a natural number based on conditions given in the problem)
- [tex]s_4\geq 5[/tex] assuming no repeats - 4 doesn't work because it is even and 1, 2, and 3 are already taken
- [tex]s_4+1+3=\text{prime}[/tex]: With three numbers, either one or three of them need to be odd to achieve a prime sum. Since 2 is the only even prime number, it follows that [tex]s_4+1+3[/tex] must be a prime number to meet the "three numbers sum to a prime number" condition
With 1, 2, 3, we can already meet one of the conditions:
[tex]\checkmark[/tex]Two numbers sum to a prime number: [tex]2+1=3=\text{prime}[/tex]
The fourth number chosen will have no effect on this and therefore we don't need to worry about this condition.
Of the three numbers 1, 2, 3, two of them are odd, so we cannot achieve a prime sum with the sum of all three. Thus, the fourth number, when added to 1 and 3, must be a prime number.
We only need to test the numbers in the set [tex]\{5, 7, 9, 11...\}[/tex] based on the conditions we found above. For each number [tex]n[/tex] you test, run the following tests:
[tex]\begin{cases}1+2+3+n=\text{prime}\\1+3+n=\text{prime}\end{cases}[/tex]
Both conditions need to be true. Testing 5, we see that 1+3+5=9, which is not prime, so 5 doesn't work.
Testing 7:
[tex]1+2+3+7=13=\text{prime}\checkmark\\1+3+7=11=\text{prime}\checkmark[/tex]
7 is the smallest value of [tex]s_4[/tex] that works and thus our set is [tex]s=\{1, 2, 3, 7\}[/tex], so[tex]\sum s_i=\boxed{13}[/tex].
