hmm whilst studying to abstract algebra today, I revisited Goldbach conjecture since its important imho. 

Trying to prove it again. I dont know if my previous proof were correct.but somehow it seems fun to try this proof tonight since is an easy type math task and crucial regarding permutation group properties regardung multiplication operation of unit groups, even not exactly same prime number concept but still hese seems as the basis to be investigated permutation groups structure of. the unit group of any number is actually a permutation in the sense of any unit number set is even numbered counted and also, each one has a counter part value of (nm -1)  = ab where a and b are both from unit number groups. So investigating structures of this permutation group for the unit group of n might give some structures in this groups and related multiplication composition operator and its relatedness to plus operator. 

i know my ai would investigate later this faster than me but i wish to have fun of this investigation of this type structures right now. maybe it turns out a not easy to find figure out structure. currently at least permutation group has some basic structure properties but needs to be investigated but at least we know that group compostiion operator has a permuation group property.      which might enable faster calculation if we know these group structures in general, that multiplicating or dividing numbers without having have to multiplicate or divide to state clearly type group behavior based properties might or might not be found/figured out. would it serve anything other than optimizing calculations? hmm might create a new basis to do calculations in that sense if such utilizable structures existed. that we might have possible power of intiuition to how very complex dynamic systems might behave more easily. this might this permutation groups might even work out in matrix type formulations since they are again even in vector form following similar group properties as expected in such cross product operation imho. excluding noninvertible matrices but i mean this basis in integers might serve good purpose in any computation problem. but there might not be, one might not find such structures easily. maybe even does not exist. but I want to investigate with also revisitng Goldbach conjecture solving attempt tonight. hmm so back to basic abstract algebra today with fun of attempting to prove the one of the most basic conjectures, of Goldbach conjecture. not as challenging as other conjectures or is it? at least nice for to try to investigate group structures there that might be very useful even in any computation even including matrices composition operators group stuctures.  which might even give insight to infinite length having matrix concepts. so even it seems as a basic abstract algebra topic this group structure topic investigatons for Goldbach conjecture are actually very fundemantal type abstract algebra concepts imho. even seems trivial conjecture at first(Whislt its not), it actually might hinder very useful group structure findings afterall. though one cant deny that this conjecture is one that is seemingly very trivial wtr to other conjecture types i saw and had not learnt alot yet. 

so tonight is attempting not very hopeful conjecture solving tries fun time.

so its fun of conjecture solving/group structures investigation fun tonight yepp. 

hmm one does think of even without ai yet, one could create some program to assist this solution attempts of Goldbach conjecture.  

yupp why not. lets go some software assisted structure analysis now. but first need to investigate alittle more with paper pencil to investigate common structure patterns myself but then would switch to checkiing structures of various examples to propgrammatically investigate some possible structure definitons. 

these unit grouos also gives hints for even matrix operations for groups of matrix with mutliplicative compostion operator. its why they call it abstract algebra. its rather basis behavior applicable to every group with similar group behaviors. i mean its easier to find some group structures from these fundemantal structures.  which also works in more nary items or so. it does not matter one can find its resemblence to another group to find its group properties. 

ok so as visible this night i am continueing revising abstract algebra. to find some structure in some very fundemental groups that might be applicable to other groups other operators. abstract algebra behaviors are universal, works in any case type behaviors it is. that is why its so cool imho. some common behavior patterns structures applicable to other groups. (with isomorphisms later i guess so). 

its like the patterns of universe is what abstract algebra is like. or common behavioral patterns applicable to other groups. 

lets continue to studying after definign tonight's very severely fun task of attempting solving Goldbach conjecture with some structure investigations. 

hmm deciding to assist with software this study (i mean structures investigations). yupp lets see.