Odd Primes

[Wictorian]

Can all odd primes be defined as the sum of a prime and a non-prime?

[Needfuldoer]

Which prime and non-prime add up to five?

[Wictorian]

17 minutes ago, Needfuldoer said:

Which prime and non-prime add up to five?

We can replace prime with 1, then 4+1= 5.

[Needfuldoer]

2 minutes ago, Wictorian said:

We can replace prime with 1, then 4+1= 5.

 

1 is not a prime number. That's adding two non-primes.

[Wictorian]

3 hours ago, Needfuldoer said:

 

1 is not a prime number. That's adding two non-primes.

I know what I did is to modify the rule to sum of a prime number or 1 and a non prime.

[tikker]

You can use that

1. all prime numbers greater than 2 have to be odd, otherwise they would have 2 as a factor, making them non-prime.
2. the sum of two odd numbers is even.
3. the sum of two even numbers is even.

to deduce that all primes >2 being odd, the non-prime to be added thus must be even (as odd+odd=even and thus non-prime). In that case the question generalises to if you can write every prime as a prime plus an even number, in which case the answer turns out to be no: https://math.stackexchange.com/questions/269658/whether-a-prime-number-can-be-written-as-sum-of-a-prime-number-and-2n?rq=1

[Wictorian]

10 hours ago, tikker said:

You can use that

1. all prime numbers greater than 2 have to be odd, otherwise they would have 2 as a factor, making them non-prime.
2. the sum of two odd numbers is even.
3. the sum of two even numbers is even.

to deduce that all primes >2 being odd, the non-prime to be added thus must be even (as odd+odd=even and thus non-prime). In that case the question generalises to if you can write every prime as a prime plus an even number, in which case the answer turns out to be no: https://math.stackexchange.com/questions/269658/whether-a-prime-number-can-be-written-as-sum-of-a-prime-number-and-2n?rq=1

The link talks about a prime plus a power of 2. I’m pretty sure that a prime can always be defined as a sum of another prime and a multple of 2 or 1.
 

[tikker]

7 hours ago, Wictorian said:

The link talks about a prime plus a power of 2. I’m pretty sure that a prime can always be defined as a sum of another prime and a multple of 2 or 1.
 

It could be, but you'll have to mathematically prove it. Multiple of 1 is out of the question I think, because everything is a multiple of 1 and the number has to be even since primes are odd. Therefore your multiple of 1 would automatically have to be a multiple of 2. That StackExchange question indeed looked at 2^n, your question would be whether any prime can be written as P2 = P1 + 2n

[Wictorian]

18 hours ago, tikker said:

It could be, but you'll have to mathematically prove it. Multiple of 1 is out of the question I think, because everything is a multiple of 1 and the number has to be even since primes are odd. Therefore your multiple of 1 would automatically have to be a multiple of 2. That StackExchange question indeed looked at 2^n, your question would be whether any prime can be written as P2 = P1 + 2n

I meant every prime can be written either as a multiple of two and a prime or the sum of a prime and 1 and a multiple of 2.

[tikker]

3 hours ago, Wictorian said:

I meant every prime can be written either as a multiple of two and a prime or the sum of a prime and 1 and a multiple of 2.

You're constantly changing the goal, now you're at a sum of three numbers. The point remains unchanged: it'll have to be proven mathematically and that may not be easy, but you'll first have to choose something you want to prove and stick with that. Since you generally cannot prove such a statement true, you will want to prove it false. In other words, you want to find an example that shows you can't write them as you propose. If you can't find such an example with reasonable certainty then you can decide to make it a conjecture (=assumed true, but not sure).