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

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 $b$?

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.
• I get it! You can only get prime numbers with consecutive $n$ only if $a, b$ are natural numbers. Thank you = ) Aug 5, 2015 at 21:47