prime numbers pattern - KYT's conjecture

[Bob_kyt]

Hello, I am Bob, a junior high school student from Taiwan.

I did some research on the disassembly of prime numbers for months and Made my own assumptions - KYT's conjecture.

More explanation about KYT's conjecture is described as follows:

a is a positive integer and is even, a>=8, b=a+18, a=c+D, c, D,E are prime numbers.

a=c+D
b=c+E

E=D+18=b-c

The even numbers a and b are split into two prime numbers. There must be a prime number c.

More examples are as follows:

10=3+7;5+5
28=5+23;11+17
46=3+43;5+41;17+29;23+23
64=3+61;5+59;11+53 ;17+47;23+41
82=3+79;11+71 ;23+59;29+53;41+41
.......
.......
10000= 4517+5483
10018= 4517+5501
10036= 4517+5519

Larger even number, it is difficult to break down into 2 prime numbers.
Is there an example where the number c does not exist?

Thanks for the help / Bob

 

[Bob_kyt]

The even numbers a and b are split into two prime numbers. There must be a prime number c.
----->
There must be a prime number c that satisfies the two equations above.

 

[Bob_kyt]

KYT's conjecture seems could be rewritten as below. - this one is better.

a is a positive integer and is even, a>=8, b=a+6, c, D,E are prime numbers.
a=c+D
b=c+E
E=D+6=b-c

There must be a prime number c that satisfies the two equations above.

 

[Dr.Peterson]

I think your conjecture is this:

Given any positive even integer a >= 8, let b = a + 6. Then there exist prime numbers C, D, and E such that a = C + D and b = C + E.​

Your third line can be derived from the other two, so it doesn't need to be part of the conjecture. But it is appropriate to focus on such derived facts. I myself would eliminate b, just noting that if a = C + D and a + 6 = C + E, then E = D + 6. So you claim to always have three primes, such that the sum of two of them is a and one differs from the third by 6.

You are asking for a counterexample. That is, you want to find an integer a that either can't be written as a sum of two primes C and D, or for which D + 6 is never prime.

Have you noticed that your conjecture depends on the truth of Goldbach's conjecture?

My own impression is that it will indeed be hard to find a counterexample, because there as numbers get larger there are usually many ways to make a sum of two primes, and many of those will differ by 6 from another prime. I would find or make a list of Goldbach sums, focus on numbers with only a few such sums, and look for the right kind of primes.

But conjectures are easy. The challenge is to prove them. Since Goldbach's isn't proved, you won't be able to prove yours (without learning a lot more, at least!). If I were you, I would put my time into learning about proofs in number theory, if that interests you.