I’m enjoying a slow morning in Tacoma, Washington waiting for my friend to get married. What better time than now to do some math? For several years now I’ve been obsessed with Goldbach’s conjecture. I’d like to share my progress on it with you.
To those who don’t know Goldbach’s conjecture is a 275 year old math problem that still has not been solved. Legend has it that it originates from German mathematician Christian Goldbach writing a letter to Leonhard Euler on June 7, 1742, in which he proposed the following conjecture:
“Every integer greater than 2 can be written as the sum of three primes.”
Euler replied by letter on June 30, 1742, and reminded Goldbach that the two had an earlier conversation where Goldbach proposed every even integer greater than 2 could be written as the sum of two primes. It is this later conjecture that has defied proof.
My progress towards proving the conjecture can be seen in the attached paper. As you can see, my approach relies heavily on accommodation, my new favorite method for dealing with problems involving prime numbers.
Enjoy and let me know your thoughts!
EDIT: I’ve received a few comments on this paper, and there are a few items that need to be addressed for clarity:
First, at the point where I’m using “El Bachroui’s Inequality,” the term we are bounding is p(π(x)+1), not p(π(x+1)). I apologize for the typo.
Second, where we are actually using anchor accommodation to prove Goldbach’s Conjecture for certain cases, some of you noted that p(π(x)-1)*p(π(x)-1) cannot be used so long as p(π(x)-1)>3. This is absolutely correct. As such, in Example 3, the accommodation picture becomes:
(3)(3) ( )( ) ( )( )
3 5 7
Likewise, Example 4 becomes:
( )( ) (3)(3) ( )( ) (3)(5)
3 5 7 11
This result actually makes it even easier to prove Goldbach’s Conjecture for these cases, as we are assured of there being at least 2 primes in Q(x), not just 1.
Thanks again for reading. I’m really enjoying the math world’s support. Happy problem solving!