I think i solved (not sure) Goldbach conjecture, but this needs several presumed lemmas that I figured out:

Lemma 1: take n and then the unit group (primes relative to n) must have some l constant that enables for every a b in the unit group of n: 

nl - ab = 1 

why?

nl -1 = ab

nl -1 is undivisible by any of C2 therefore from C1 and there by ab also must be from C1 

and reversely take a,b from C1 and do apply this logic it turns out there must be an l that nl-ab = 1 

so there always exists an integer l which enable -> nl -ab =1 for every n (and it's C1/C2 groups and a, b from being C1.)

 

Lemma2 : 

as everyone knows, the unit group is  cyclic (no i need to revise abstract algebra it werenot called cyclic this term but would write when after i revise ) that for every c in unit group there is a, b that creates ->

 a*b = c

Now coming to solution:

For every n there are 2 class, one class being unit group the other being the other nonunit group

thereby: 

first class consists of primes that are not factors of n inclouding 1 even 1 not being prime its considered in unit group and their factorizations

second class consists of primes and primes factorizations tht are having a common denominator with n

e.g. prime numbers of factoriation of n and their other possible factorizations from inside the class

now lets call Class 1 as C1 and class 2 as C2 of this n: 

now lets make some change to variables,

now as Goldbach conjecture says every even number is sum of two prime numbers:

lets call this n as 2n from this part. 

so if this number 2n were sum of 2 parts first thing to think of its constitutents would be:

n + n right? 

(n is C2 as is a factor of 2n.)

Now:

lets create two numbers:

a  = n + m

b = n - m

our proof would show that a and b are two prime numbers from C1.

so first thing to prove is: 

does such a, b exist (and not needing to be unique or not that is not relevant) that can create such condition. 

hmm lets first check from: 

a * b = n_square - m_square right?

so if we rerepresent a b as:

a = n + m 

b = n -  m

lets take

p1 p2 p3 as first second third ... relative prime number (C1 number) coming before n 

e.g.

p3 .. p2 .. p1 n 

for this p1 ->  if p1 is not prime, go back and find the p which is prime e.g. p2

now with the found p num:  lets recall it p1: 

p1  = n - m

so m is n - p1 and b is then 2n-p1

so we should prove that 2n - p1 exists and is prime. 

so b is: (2n - p1 is)

2*n - p1 * (2*n*x - p1)       where x is number which makes 2*n*x - p1  =1  we definitely know suych number exists from lemma 1

now take this number: 

(2*n  - 2* n* x* p1 )+ p1_square

tthen the equation becomes   (2*n  - 2* n* x* p1 )+ p1_square

hmm just a minute maybe its not proved :D

this number is visibly not visible by 2n  --> so its from C1 now lets prove its prime (nonrelative prime)

take any number from C1 it cant divide this. 

take any number from C2 it cant divide this. 

so its proved. even I could not write in formal maths language yet since i am not in to proofs. but its visibly proven. 

so b also exists. there by Goldbach conjecture is true, there exists a and b which are primes and for which a + b = 2n 

 (maybe there is some error in proof and i am not seeing it thats also possible. so need to rework on this proof with unit tests later)

I guess this one were one very easy to prove conjectures, some kind of trivial or maybe most trivial conjecture to prove.

I am going to invest time more to these number properties to investigate its group structures.  to figure out structures which might be used in optimizations of any calculation. there might not be such structures but still worth exploring this effort since I just am curious to learning if there is such group structures there. then it could be used in isomorphisms to do other optimizations in other groups. so I very find it very important this unit group concept and its group operator * and its as a group oeprator +  group type etc.  

 curious guh. I just do this study due to curiosity. I instead normally need to continue ai project and continue abstract algebra topology study but due to curioisity I just wonder this group structure since if there is it might mean finding new optimizations for division/multiplication including topics even seems like most possibly there does not exist such structures most possibly but lets see. i am curious to investigating this groups structure. since integers also should have a ontology def in maths grammar and better to investigate this topic more before moving to maths grammar study of ontology definitions imho. 

now that I proven Goldbach I can have nice happy rest for rest of day. I were fixated on finding a proof for this very trivial conjecture and would not have peace  until i figure out so now that its proven i can rest happily afterwards. since tihs structure studies are very important solving initially Goldbach conjecture were very important but this is juhst the beginning of this structure study which i wish i could finalize in 3 or 4 days or so. then can continue revising abstract algebra again. 

-----------------------------------------------------------------------------------------------------------------

I have to prove add proof of lemma 1 of course and lemma 2. lets add to an article lets write an article for this and prove lemma 1 and lemma 2 also to there.

---------------------------------