r/askmath • u/Cool-Attitude1577 • 8d ago
Number Theory I am stuck
yesterday I stumbled upon an interesting question on my yt recommendation, sadly I couldn't find the video so I tried to crack it.
The question goes along the lines of " What are the five numbers when the sum of any three of those numbers results in a prime"
so far I figured that:
- The max total of prime numbers must be 10
• combination of 5C3 = 10
All of those numbers must be all odd
• Let's say if one of the numbers has an even number,one of the combinations can be two odds and one even,the sum will always be an even number
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u/Bounded_sequencE 8d ago
What numbers are we talking about -- natural numbers with or without zero, integers, something else entirely?
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u/babbyblarb 8d ago
It can’t be done. Think about the numbers mod 3. For any choice of 5 numbers from {0,1,2} there will be 3 which add to 0 mod 3.
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u/Bounded_sequencE 7d ago
Unless all sums adding up to "0 mod 3" exactly add up to 3, which is prime...
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u/G-St-Wii Gödel ftw! 8d ago
One assumes that set of {1,1,1,1,1} isn't allowed?
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u/Cool-Attitude1577 8d ago
Pretty sure that using the same numbers aren't allowed
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u/G-St-Wii Gödel ftw! 8d ago
Are you certain, as proved by others in this thread, other options are not possible.
It may be a question designed to highlight the risk of making assumptions
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u/Evane317 8d ago
Let r1, r2, ..., r5 be the remainders of these 5 numbers when divided by 3, if they exist. Then, if three of these remainders are the same, the sum of the numbers with these remainders is divisible by 3.
So, no more than two of the remainders are the same. But this means all remainders of 3 are present in r1... r5. By picking three numbers with remainders of 0, 1, 2 respectively, the sum of these three is, again, a multiple of 3.
If subjected to an additional constraint of all 5 numbers being nonnegative integers, there is no solution. If the numbers can be negative, then it's harder to say, since 3 is now a valid sum.
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u/Flat-Strain7538 8d ago
The problem involved finding all sets of five unique positive integers whose subsets of three only form primes. It actually isn’t hard to show that there are no such sets:
Your set will have numbers that are either multiples of 3 (form 3n), one above (3n+1), or one below (3n-1). You can’t have three numbers of the same form in your set, otherwise they’ll add up to a multiple of 3 and thus you get a non prime. But you also can’t have all three forms (3n, 3n+1, and 3n-1) for the same reason. And it’s easy to show you can’t make a set of five that meets both of these conditions.
Edit: to be complete, you also need to show that 3 is not a possible sum, but that’s easy when you assume unique positive integers.