Question

1. subst. - let p and k , two prime numbers greater or equal 2,from the set of prime numbers, P, in the form : p=2a + 1 and k=2b + 1 , such that a and b are natural numbers,from the set of natural numbers N, - let m=2n ,m greater or equal 4,even number,from the set of natural numbers N and n grater or equal 2,natural number from set of natural numbers N, 2. concl. - every even integer greater than 2 can be expressed as the sum of two primes 3. prove:- by ,,reductio ad absurdum” * - step 0. for n=2 --- m=4 --- 4=2+2 ** - step 1. for - if n is greater or equal 3 so always will be a number a and

Answer 1

1. subst. - let p and k , two prime numbers greater or equal 2,from the set of prime numbers, P, in the form : p=2a + 1 and k=2b + 1 , such that a and b are natural numbers,from the set of natural numbers N, - let m=2n ,m greater or equal 4,even number,from the set of natural numbers N and n grater or equal 2,natural number from set of natural numbers N, 2. concl. - every even integer greater than 2 can be expressed as the sum of two primes 3. prove:- by ,,reductio ad absurdum” * - step 0. for n=2 --- m=4 --- 4=2+2 ** - step 1. for - if n is greater or equal 3 so always will be a number a and b such that n=a + b + 1 - prove. 3=1+1+1 4=2+1+1 5=2 +2+1 ................ n=a+b+ 1 - so for n=k than k=a+b+1 - suppose that is true - for k+1=(a+b+1)+1=(k)+1=k+1 - so for k+1 is true - for n grater than 2 always will be a number a and b such that n =a+b+ 1 *** - step 2. - every even integer greater than 4 can be expressed as the sum of two primes m=p+k - prove by ,,reductio ad absurdum" - so than m is not equal p+k - so than 2n is not equal 2a+1+2b+1 - so than 2n is not equal 2a+2b+2 / divide both sides by 2 - so than n is not equal a+b+1 - so what is in contradiction with the proof from step 1. where n=a+b+1 was proved that is true - so than m=p+k is proved that is true so,, every even integer greater than 4 can be expressed as the sum of two primes” --- q.e.d.

Answer 2

Please properly format it...

Answer 3

1. wann being substitutions

2. conclusion -- resulte - deduction

3. proof. of

Answer 4

so chek it please there on this site :

https://docs.google.com/document/d/1P26hfFM6OCJONXQJ3I4T6bqgNcfce06srIR7mrb1h9U/edit

opinions please !!!

Answer 5

It looks to be a valid proof. But, you may want to clear up your notation a bit:

It's convention that p and q are reserved for prime numbers. I would use those instead of p and k.

When doing an induction proof, you do not assume true for n. When you say n = a +b + 1, you are saying this holds for all n. You don't know that. That's what you're trying to prove. So, after proving the initial case, you assume it holds true for some fixed k which is in n. Then, you show that the k+1 step holds because k can be substituted into the equation (everyone makes this mistake, including me).

So, I would use k instead of n (because you've already defined n as the entire set, not a fixed number in that set). Further, I would elaborate more on the k +1 step so that you have: k+1 = (k)+1 = (a+b+1)+1 = a + (b+1) + 1. It's essentially the same, but you're adding more clarity that this can be done by using k.

Otherwise, it looks pretty good. Does your teacher/professor force you to break it into three parts (1,2, and 3) or is that out of your own habit? I don't see that where I'm from, but things could be done differently in other parts of the world.

Answer 6

@DanielxAK first of all thank you so much for your reply,opinion from this my ,,proof"

...so can you please clarifie some from your words,expressions from your reply for i can ending it acceptably - hope so much for ALL interested people - because how you have wrote here and how i have understood it from your words you see that can being this my proof one ,,good" , understandably -- acceptably - ,,interesting" proof of ,,Goldbach's conjecture"

- wait so much your reply from this and wann please you again chek,look,see my proof from ,,Collatz conjecture" too please and write me please what is your opinion from this too
- here is the link where you can read it easy ,clearly :

https://docs.google.com/document/d/1q2DpLVjAvGSjJCs6bme04Nlr6KmZqvS16bdRc1FpnSY/edit?pli=1

thank you for your all effort ,opinions and cooperation in this cases

wait your reply soon and hope so much that you will can understanding this ,,Collatz's conjecture" proof too the same easy like the ,,Goldbach's conjecture " proof

good luck and Happy weekend

jhonyy9

Answer 7

@sauravshakya hi ! so what is your opinion from this proof ? can be this understanding right easy or i need clarifie ,again some points,words ,expresions ,..., ? what is your opinion ,please ?

Answer 8

so do you understand it right ,easy ,correct till the end ?

Answer 9

PLZ WAIT A MINUTE

Answer 10

I think there is a problem:
n=a+b+1 (THIS IS ALRIGHT)
But: a and b are not all natural numbers....... for example when a=4 , p=2*4+1=9
Where p is not prime when a =4.....
So, how can n be all natural numbers.

Answer 11

For example: when n=12=4+7+1

Answer 12

There: a=4 and b=7
Now, p=9 and k=15
???????

Answer 13

@mukushla could help..... But he is offline.

Answer 14

so thank you for your cooperation and from this your words result that my proof ,words not can being understood it all right,correct ,easy

- so because there i wann trying to prove that every prims can be writing in the form of 2n +1 -- by definition ,so is true sure --- and from this result that if p and q or k prims so always will be an a and a b natural numbers for p and q and k prims when we can writing p=2a+1 or q=2b+1 or k=2c+1
- this is resulted from the definition of prims graters than 2
hope so much that this will clarifie anything from what was ununderstandably there ...

ok ?

Answer 15

oh i see it now so firstly was proved that every natural numbers graters than 2 can be writing in the form of a+b+1 so n=a+b+1

so what is naturaly sure because by definition we know that the line of natural numbers greating - i think that not is this word the right word here - so that the line of natural numbers greating by 1 to 1 so than from this reaulted that every natural numbers greater than 2 wil be possibil writing in the form of a+b+1

right ?

Answer 16

so and this not is in contact with prims there again

Answer 17

so this is just a first introduction for i can proving using this after that by reductio ad absurdum the Goldbach' conjecture

hope so much that now is understandably right,correct sure