Goldbach's Conjecture

[science_man_88]

Originally Posted by CRGreathouse

Can you prove that this always happens -- that for large enough n, there is a prime p < n such that 2n - p is composite?

I think this is a misunderstanding 15 is equidistant from 13 and 17 but it's not prime is what i'm saying so we need rules to stop finding solutions like this. the rules that normally slow down prime searching work but then what's the point of the prime equidistant from 2 primes part since it wouldn't help that.

[3.14159]

well if it shows all primes under it could we then find all primes under a given even number ? try it with 164200 for me.

You mean, if all primes eventually decompose into sums of 2s and 3s? Yes.

164200 = 23039 + 141161

Decomposing it into sums of 2s and 3s is easy:

23039 = 2(5548) + 3(2648) + 2(1523) + 3(311) + 2(10)
141161 = 2(34267) + 3(20264) + 2(4267) + 3(1068) + 2(20) + 3(5) + 2(1)

The decomposition of 164200 into 2s and 3s: 2(5548) + 3(2648) + 2(1523) + 3(311) + 2(10) + 2(34267) + 3(20264) + 2(4267) + 3(1068) + 2(20) + 3(5) + 2(1).

To check if it adds up: Dammit, 40 short. Checking sums... I see the problem:

Here's the right decomposition.
2(5548) + 3(2648) + 2(1523) + 3(311) + 2(10) + 2(34267) + 3(20264) + 2(4267) + 3(1068) + 2(40) + 3(5) + 2(1).

Decomposing primes into sums of 2s and 3s:
2: 2
3: 3
5: 2+3
7: 2+2+3
11: 3+3+2
13: 3+3+2+2
17: 3(5) + 2
19: 3(5) + 2(2)
23: 3(7) + 2
29: 3(7) + 2(4)
31: 3(9) + 2(2)
37: 2(14) + 3(3)
41: 2(16) + 3(3)
43: 3(11) + 2(5)
47: 2(19) + 3(3)
By the way, is this silly conjecture proven? (Every prime can be written as a sum of 2s and 3s.) And, who thought of it prior to me? I stumbled upon this quite a while ago.
I think I can generalize it to: Every integer greater than 3 can be written as a sum of 2s and 3s.

[science_man_88]

Originally Posted by 3.14159

You mean, if all primes eventually decompose into sums of 2s and 3s? Yes.

164200 = 23039 + 141161

Decomposing it into sums of 2s and 3s is easy:

23039 = 2(5548) + 3(2648) + 2(1523) + 3(311) + 2(10)
141161 = 2(34267) + 3(20264) + 2(4267) + 3(1068) + 2(20) + 3(5) + 2(1)

The decomposition of 164200 into 2s and 3s: 2(5548) + 3(2648) + 2(1523) + 3(311) + 2(10) + 2(34267) + 3(20264) + 2(4267) + 3(1068) + 2(20) + 3(5) + 2(1).

To check if it adds up: Dammit, 40 short. Checking sums... I see the problem:

Here's the right decomposition.
2(5548) + 3(2648) + 2(1523) + 3(311) + 2(10) + 2(34267) + 3(20264) + 2(4267) + 3(1068) + 2(40) + 3(5) + 2(1).

I mean 164200 = 23039 + 141161

23039 = 10+23029(prime)
10 = 7+3
141161= 4+1411573(prime)
4=2+2

so in a way yes except finding more primes under the number along the way.

even if you have a od dinstead of a prime as long as it can be brought down to a even number idf Goldbachs conjecture is right you could eventually find most primes by turning a row even then prime then a even odd then even then prime again etc. trying to get most primes out.

[CRGreathouse]

Originally Posted by 3.14159

By the way, is this silly conjecture proven? (Every prime can be written as a sum of 2s and 3s.) And, who thought of it prior to me? I stumbled upon this quite a while ago.
I think I can generalize it to: Every integer greater than 3 can be written as a sum of 2s and 3s.

Sylvester had a general solution to the problem of deciding the least integer N(A, B) such that all n > N could be represented as aA + bB. Your particular case was probably known to the ancients.

[3.14159]

Sylvester had a general solution to the problem of deciding the least integer N(A, B) such that all n > N could be represented as aA + bB. Your particular case was probably known to the ancients.

The ancients?

[CRGreathouse]

YA RLY.

2 and 3 can be represented in that form, so for any n >= 4 just take the representation of n-2 and add a two. If you want at least one of each, start at 7 and 8 (and check 2 through 6). None of that was out of reach of the ancient Greeks or even Egyptians.

[science_man_88]

I know this is kinda out there but since prime + even = odd(or prime a subtype almost) and

prime + odd = even is it fair to say that if a number can't be expressed as the sum of 2 primes it can't be represented with even or odd numbers as well hence it should not exist ?

edit: primes>=3

[CRGreathouse]

Originally Posted by science_man_88

prime + odd = even is it fair to say that if a number can't be expressed as the sum of 2 primes it can't be represented with even or odd numbers as well hence it should not exist ?

edit: primes>=3

No. The potential counterexample to Goldbach's conjecture is a number n such that for p = 3, 5, 7, 11, ..., precprime(n), 2n - p is composite. We don't think there is such a number, but parity considerations don't rule it out.

[science_man_88]

Originally Posted by CRGreathouse

No. The potential counterexample to Goldbach's conjecture is a number n such that for p = 3, 5, 7, 11, ..., precprime(n), 2n - p is composite. We don't think there is such a number, but parity considerations don't rule it out.

I know nothing about that lol.

please explain what the added 2n-p means ? I ran precprime for n = 1,20 and If i'm right it's the most recent prime found. what does the 2n-p mean ?

[CRGreathouse]

Originally Posted by science_man_88

what does the 2n-p mean ?

?

n is a positive integer, p is a prime number, and 2n-p is two times n minus p.

precprime(n) is the largest prime <= n.

[science_man_88]

Originally Posted by CRGreathouse

?

n is a positive integer, p is a prime number, and 2n-p is two times n minus p.

precprime(n) is the largest prime <= n.

Okay good information. If n is half c then it becomes c-n(the n in my equation) which should come to the top of the 2 possible primes in my equations(according to your code).