Theory behind prime generating function $p=an+b$, where $a, b$ are real coefficients

Question

The Green–Tao theorem states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words there exist arithmetic progressions of primes, with k terms, where k can be any natural number.

For example

$$ p = 43142746595714191 + 23681770 \times 223092870 \times n \\ n \in [0, 1, 2, ..., 25]: p \in \mathbb{P} \tag{1} $$

where $ \mathbb{P} $ is the set of prime numbers.

If you consider the general formula

$$ p = a \times n + b \tag{2} $$

where $ a, b \in \mathbb{R} $, $ n \in \mathbb{N} $, and $ p \in \mathbb{P} $, you can also generate primes for arbitrary coefficients $ a $, and $ b $. However, I can't seem to find an example that will generate prime numbers for consecutive $ n $, like in (1).

Is there a theory or conjecture which talks about (2) with real number coefficients $ a $, and $ b $?

Answer 1

If $f(n) = an+b$ is an integer for two consecutive integers $n=n_1$ and $n=n_1+1$, then $a = f(n_1+1) - f(n_1)$ and $b = f(n_1) - a n_1$ are integers. So no extra generality is obtained by considering $a$ and $b$ to be real numbers rather than integers.