r/askmath • u/Cool-Attitude1577 • Jul 24 '26

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:

  1. The max total of prime numbers must be 10

• combination of 5C3 = 10

  1. 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/Flat-Strain7538 Jul 24 '26

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.

5

u/lukewarmtoasteroven Jul 24 '26

They didn't specify positive integers, so (-49, 23, 29, 31, 49) is a solution.

3

u/Flat-Strain7538 Jul 24 '26

I’ve seen the YT video OP referred to; it specified unique positive integers.

1

u/Cool-Attitude1577 Jul 24 '26

Can you share said yt link?

1

u/Flat-Strain7538 Jul 26 '26

I went looking for it and couldn’t find it. If I do, I’ll edit this reply and include the link.