Is the following array of numbers related to Goldbach's conjecture?

Question

1*1
1*3 2*2 3*1
1*5 2*4 3*3 4*2 5*1
1*7 2*6 3*5 4*4 5*3 6*2 7*1
*
*
*
etc.

If we go on constructing the pattern of numbers above, will each row contain atleast 1 product of two odd prime numbers? If the answer to this can be proven to be yes, then would this prove also Goldbach’s conjecture considering the fact that the sum of any two numbers composing a product in a row adds up to twice the square root of the perfect square contained in that row, and that all even numbers greater then 4 are accounted for in the matrix?

For example, in the row that contains the perfect square 2*2, we have 3+1 = 4, with 3 and 1 odd prime numbers. In the row containing the perfect square 3*3 we have 5+1 = 6, with 5 and 1 odd prime numbers…etc.

Might this be related to Goldbach’s conjecture?

Thanks very good points! I, indeed, overlooked the fact that 1 is not prime.
The way to generate the rows is exactly as you said, the nth row conists of all p*q that satisfy p+q = 2n, where n greater than or equal to 1. I gather then that it is then just a restatement of goldbach's conjecture in a different form. Isn't there a theorem that states that there must exist a prime number between n and 2n, where n is some positive integer greater than 1? That probably doesn't help either, does it?

Answer 1

You didn't give an actual rule for how you are making these rows, here is how I'm reading it:

The nth row will consist of all p*q such that p+q = 2*n. p,q,n in the set of Natural Numbers.

If you are using this then what you are saying is "if we prove G's conjecture, then we prove G's conjecture." You just restated it in a different form and restricted yourself as to which prime number's you are using in your sum to get the nth even number.

To answer your question "might this be related to Goldbach's conjecture?" then yes.

Good thinking though.

P.S. Add details if I misread how you were forming those rows, it's a little hard to guess without being explicitly given the rule.

Edit:
Here are some sites with a bunch of properties of primes on them.
http://mathworld.wolfram.com/PrimeNumber.html
http://en.wikipedia.org/wiki/Prime_number

I think what you were talking about in your additional detail was Polignac's conjecture (it's on the wikipedia page).

Goldbach's conjecture has had multi-million dollar prizes offered before for its proof, so it isn't going to yield to any easy tricks. That shouldn't stop you from trying but it'll be an intense effort with little chance of success. Still, there's a first time for everything. Good luck!

Answer 2

Since when is 1 considered a prime number? (its not)

In fact Goldbachs conjecture http://mathworld.wolfram.com/GoldbachConjecture.html

speaks about numbers > 4

So let us consider only those.

Yes the pattern here is equivalent to goldbachs conjecture because the ith even number corresponds to the sum of the numbers in the ith row. But proving that each row contains the product of two primes is exactly what you need to do.

But i dont see how this will help

Answer 3

Cool idea. One is not a prime, by the way.