Question

given p and q two primes greater than 2
P=2a+1 and q=2b+1 with a and b naturals
prove that for n > 2
n=a+b+1 is always true
- what proof method may be more usefully by contradiction or math.induction or ... ?

Answer 1

@ganeshie8
@mathmate
@FaiqRaees
@TheSmartOne
@Directrix
@Kainui

Answer 2

@DrMoss :(

Answer 3

What is n?

Answer 4

n is natural greater than 2 wrote there

Answer 5

for all n?

Answer 6

yes for all n greater than 2

Answer 7

if p =3, q =5, then p =2*1 +1, q = 2*2 +1, hence a =1,b = 2 , then n = 1+2 +1 =4
That is the least of prime p, q >2, therefore with n=3, how can you have the proof valid?

Answer 8

n is always odd>3, that makes more sense.
To me,
p = 2a+1, then \(a=\dfrac{p-1}{2}\)
q=2b+1, then \(b=\dfrac{q-1}{2}\)
Let \(n = a+b+1=\dfrac{p-1}{2}+\dfrac{q-1}{2}+1=\dfrac{p-1+q-1+2}{2}=\dfrac{p+q}{2}\)

Answer 9

If Goldbach's conjecture were accepted as true, it will take care of all the odd naturals >2. In other words, if you can prove the above, you have proved Goldbach's conjecture.

Answer 10

@Loser66 for n=3 this is valid really - look please

there are p,q and n greater than 2

p=2a+1 and q=2b+1

so a = (p-1)/2 and b=(q-1)/2 - yes ?

than n=a +b +1 so for p=3 =>a=3-1/2 =>a=1
for q=3 =>b=(3-1)/2 =>b=1

that we have n= 1+1+1 so n=3

do you believe it now that is true for n=3 ?

hope helped

Answer 11

@ganeshie8 what is your opinion please about above wrote ? thank you

Answer 12

I am confused about the question.

Given primes p and q > 2 we know they are odd so p=2a+b, q=2b+1

Now we are saying that for all n>2 we have n=a+b+1?

I am confused that it says "given p and q"

Answer 13

Agree, the question by itself it's not clear.
@jhonyy9 added the following comments.
`n is natural greater than 2 wrote there `
`(for all n?)`
`yes for all n greater than 2`
It is not clear if it was part of the question, or if it was interpreted by OP.

My previous response was based on his comments being valid.

Answer 14

@mathmate sorry what is not clearly there ? in the text of this question i wrote n>2 and n natural - so what is difficile in this - sorry but i dont understand it

hope is clearly now for everybody - ty.

Answer 15

What is not clear to me are the following:
1. As @loser66 says, `what is n?`
If a and b are naturals, than a+b+1 is always natural. Is that what you needed to prove? or
n>2 is what you need to prove?

2. You mentioned later that `yes for all n greater than 2`, is that your interpretation or is it part of the question? Why is it not mentioned in the question?

3. for the initial part of the question:
`given p and q two primes greater than 2 `
is it for the two particular given p and q, or "given p and q are \(any\) primes greater than two" ?

If the original question is in a different language, I suggest the original question be posted to avoid confusion. Translating mathematical text is very tricky.

Answer 16

*
`for **all** n?`
`yes for all n greater than 2 `

Answer 17

@mathmate sorry but this may be more clearly than i ve wrote just for n > 2 and that always exist p and q ,p=2a+1 and q=2b+1 so p,q primes greater than 2
prove that this statement n= a +b +1 is true

this may be more understandably text of above wrote exercise ? thank you