Question

- let a,b and n - numbers naturals from N,
a>=1,b>=1 and n>=3,
- how can prove it that for any values of n always will be one number a and one number b such that this equation n=a+b+1 is true ?

dem: if n>=3 so (n-1)>=2 so (n-1)/2 >=1
- if (n-1)/2 >=1 so are a and b >=1 so
n=a+b+1 will be n=(n-1)/2 +(n-1)/2 +1
- if n=3
3=(3-1)/2 +(3-1)/2 +1
3=1+1+1
3=3
- is this prove correct please ?

Answer 1

In your proof above you moved away from natural numbers.
How about something like below ?


For any natural number n>=3 , prove there exist natural numbers a>=1 and b>=1 such that:

n=a+b+1

Let m=n-1 :

For any natural number m>=2 , prove there exist natural numbers a>=1 and b>=1 such that:

m=a+b

Pick a=1 ; satisfies a>=1
Pick b=(m-1) ; satisfies b>=1 since m>=2

Then -

m = a + b = 1 + (m-1) = m

Answer 2

- can may be accepting correct this prove ?

Answer 3

what i have wrote

Answer 4

When you say "n=a+b+1" , and then say "n=(n-1)/2 +(n-1)/2 +1 "
you are saying that "a=(n-1)/2" and "b=(n-1)/2" .
When n is an odd number, it will work because "a=(n-1)/2" and "b=(n-1)/2" will be natural numbers. But when n is an even number, then your a and b will not be natural numbers.

Answer 5

You could say that for odd values of n (n>=3) , what you originally did, i.e. pick:
a=(n-1)/2 and b=(n-1)/2

And for even values of n (n>=4) , pick:
a=n/2 and b=(n-2)/2 ;
n=a+b+1 ; n=(n/2) + (n-2)/2 + 1 ; n= ( n + n - 2)/2 + 1; n=(n-1)+1 ; n=n

Answer 6

- so i have asked this in this way because a and b are grater or equal 1 and from n>=3 have resulted that (n-1)/2 >=1 too - hence i have thought that this equation can be writing like n=(n-1)/2 +(n-1)/2 +1 is it this correct ?

Answer 7

The point is that the problem asks you to prove that there are NATURAL numbers a , b, N such that a>=1 and b>=1 and n>=3 for which n=a+b+1 .
So you need to show which values of a and b will satisfy ALL the requirements (including them being natural numbers).

For odd values of n ( n = 3, 5, 7, 8, 9, ... ) your proof works, because you can pick:
a=(n-1)/2 and b=(n-1)/2

But for even values of n ( n=4, 6, 8, 10, ... ) you need to pick different values for a and b so that they will be natural numbers:

a=(n/2) and b=(n-2)/2

Answer 8

The problem is that in your original proof, if n is an even number, then a and b won't be natural numbers.

For example, when n=4 then
(n-1)/2 = (4-1)/2 = 3/2 = 1.5 which is not a natural number (a counting number).

See the definition of natural numbers here:

http://en.wikipedia.org/wiki/Natural_numbers