Lock The Gates!
As you may already know, there is this “thing” called the Goldbach Conjecture that is one of the oldest and best known unsolved problems in number theory and in all of mathematics. It was first stated more than 250 years ago and has remained unsolved ever since.
One of the most beautiful characteristics of the Goldbach Conjecture is its simplicity. Here it is:
Every even integer greater than 2
can be expressed as the sum of two primes.
Again, however, the above Goldbach Conjecture has never been proven but is generally considered to be true and has been show to hold up thru at least 4 x 1018
The above conjecture is known as the Goldbach “STRONG” Conjecture; which implies (you guessed it) that there is also a “WEAK” conjecture. They are related in that the strong conjecture implies the weak conjecture.
The weak conjecture is also a marvel of simplicity. Here it is:
Every odd number greater than 5
can be expressed as the sum of three primes
(A prime may be used more than once in the same sum).
It was initially my plan to start a “hobby” project of a C# program related to the STRONG conjecture that would………
Anyway, that “plan” took a detour…to Los Angeles and the red planet (Mars). Luck (some good and some bad), and circumstance led down a different and unexpected path. It led to developing 2 Proofs of interesting, unexpected, and rather curious features of both the strong AND weak Goldbach Conjectures. The unexpected and curious properties are even more interesting when considered together. With that in mind, they are shown below (side by side to emphasize their curious properties). For the associated Proofs and their human interest origins please read the associated articles (links are below).
Goldbach’s STRONG conjecture states:
All positive EVEN integers ε > 2 are the sum of 2 primes.
Gerry’s Goldbach Strong Theorem:
The Prime Factors of ε, are
never one of its Goldbach Primes
except for the case where ε is “constructed” by simply doubling some prime.
Goldbach’s WEAK conjecture states:
All positive ODD integers Ω > 5 are the sum of 3 primes.
Gerry’s Goldbach Weak Theorem:
The Prime Factors of Ω are
always one of its Goldbach Primes
EXCEPT FOR THE CASE where the odd number Ω is itself Prime
Open The Gates!