GOLDBACH FUNCTION

                                      GOLDBACH FUNCTION

                    

                    Goldbach Conjecture is one of the famous conjecture in mathematics from many of years, mathematicians are still try to solve this till the time of conjecture discovered. Try to solve this famous conjecture mathematicians are discovered many new things related to this conjecture some of are proved. We seen mathematicians discovered many another versions of Goldbach Conjecture like pictorial form and graph form and computer program also. Today in the time many computer programs  proved that Goldbach Conjecture is true upto a highest integer.

                    

                    Some months ago I get an opportunity to share my views on a competition about Goldbach Conjecture I make a poster for this competition so last some months I thought about Goldbach Conjecture got many interesting things about this conjecture that not only gave this conjecture another view also I belive one of easyest way to express this conjecture by the use of basic primary concept of mathematics that helps others to easyli understand this conjecture.

                   I start with what Sir Goldbach say then I will show my idea and some works related to it that its truely amazing for me I hoped everyone read this full vlog and comment me their oppinions about this and help me to do more better .....

                 Sir Goldbach in 1742 in his letter to Sir Leonhard Euler proposed about this conjecture on back Sir Euler reply him that "I regard this as a completely certain theorem, although I can't prove it". 

 

   GOLDBACH CONJECTURE :- "Every even integer can be expressed as a sum of two primes"

              

               At the time of sir Goldbach mathematicians consider 1 as a prime number so sir includes this but later on the conjecture turns to,

            "Every even integer greater gthan 2 can be expressed as a sum of two primes"

 

         Now I introduced you to a new concept Goldbach Function.

                   So, What is Goldbach Function?

 Goldbach Function:-   

            Let us consider two set, 

                A = Set of all prime numbers

                B= Set of all positive even integers (except 2)

           Let us define a function, f: (A × A) --> B  

                  by f(x,y) =x + y, where (x, y) ∈ (A × A) 

    This function is a surjective function i.e., for all m ∈ B there exist such x, y ∈ A

       

   Example- 1. Let 6 ∈ B so we get (3,3) so the function is surjective.

             2. Let 10 ∈ B so we get (5,5),(7,3) so the function is surjective.

  NOTE- There is no distinction between (x,y) and (y,x) as set of all positive integer set is commutative. 

    One realy important thing I mentioned here that is since 2 is only an even prime number its hits the definition of function as 2+(prime number except 2 ) not equals to an even integer so for that we not consider 2 as an element of A. By this you saw that 2 things vary one is which goldbach partition 2 as sum with prime give the even integer gives 1 less number of partition as second is 4 belongs to B but 4 has not any preimage so we not consider 4 as an element of B.

    

         So the interesting thing is this function is not only another version of the Goldbach Conjecture rather than is one of good example of many of properties that show below , I define some new moderation of some pre-concepts.

         

    Is the function is bijective ?

   Example 2 clearly show the function is not injective hence not bijective function.

 

Now lets define some interesting things, 

          " x ρ y if and only if x+y is an even interger "

   

         1) x + x =2x so is ρ reflexive relation.

       2) If x + y is even integer then y+x is an even integer as even integer satisfy the commutative property so is ρ symmetric relation .

      3) If x + y is even integer and  y + z is even integer then x + z is even integer so is ρ transetive relation.

        4) ρ is not antisymtric relation because (5,5) and (7,3) both give even ineger but 5 ≠ 7 and 5 ≠ 3

      Hence its an equvalence relation but not POSET.

          Although its an welldefined relation but still when you look at the function you see cartesian product of two sets gives an another set so we defined new kind of class of sets elements.

  SOUI OF CLASS :-  Soui of class denoted by S(a) where a belongs to set B for all elements of B we get an soui of class its defined by 

                    S(a)={(x,y)| x+y = a } 

  

            Now its easily seen that S(a) and S(b) for any arbitary a ,b ,

 S(a) ∩  S(b) = ϕ  and   ∪ S(a) for all a = B 

           So this gives us partition called Goldbach Partition.