Is the following little sieve for generating primes satisfying Hypothesis H valid? Known? Obvious?
Let $F$ be a set of functions satisfying Hypothesis H and let $I$ be the integers greater than 1 (2 in some cases). For each $f \in F$ let $I_j(f)$ be the integers $i$ for which $f(i) \in (I+j)/jI$. Finally, let $F_k$ be the intersection of $I_j(f)$, j=1(1)k, $f \in F$ and $p_k = 1 + \min F_k$. Then $f(p_k)$ is prime for each $f$ in $F$.
Cheers, Scott
P.S. I'm happy to send along a Mathematica notebook that implements the sieve to anybody interested.
