Question

- let p and k prims,p>=2 and k>=2,
- let a and b natural numbers ,a>=1 and b>=1,
prove that for every n>=2 exist one a=(p-1)/2 and b=(k-1)/2 such that this equation n=a+b+1 is true .

-for example :
2=(2-1)/2 +(2-1)/2 +1
2= 1/2 +1/2 +1
2=1+1
2=2
or 3=(3-1)/2 +(3-1)/2 +1
3=1+1+1
3=3
or
4=(5-1)/2 +(3-1)/2 +1
4=2+1+1
4=4

Answer 1

n = (p-1)/2 + (k-1)/2 +1 = (p+k)/2.
Thus, what the question is actually asking is to prove that are always an infinite number of primes>2 such that the sum of any two of them is even.But this is obviously true as any prime>2 is odd and the sum of two odd numbers is always even.