Can the proof of one conjecture be considered a proof of the other conjecture?
The general method of building an infinite number of prime producing quadratic polynomials was given in the link below.
Here we will give just one example of two prime producing polynomials which can generate twin primes. We start with the following polynomial that can only produce even integers and we add a "prime seed part" to it to make it generate primes.
e(n) = (10+n)(1+n) = n^2 + 11n + 10
then we add the "prime seed part" 2n+1 to it to get:
p(n) = n^2 + 13n + 11
Now to get a twin prime we need another prime producing polynomial, g(n), which differs by 2 from p(n).
g(n) = p(n) + 2 = n^2 + 13n + 13
For n=0,...,10 the combination of these two polynomials produced 5 pairs of twin primes.
So does the proof of one conjecture lead to the proof of the other conjecture?