Does the existence of a power of $2$ prime-representing function $\lfloor \alpha^{2^n} \rfloor$ imply Legendre's conjecture to be true?

Question

I am preparing a paper on the topic of prime-representing functions, and one of the referees that reviewed my paper suggested me to include references to the following state of the art article: "Prime representing functions", K. Matomäki, Acta Math. Hungar.,128,no.4 (2010) 307--314.

The abstract of the article is:

We construct prime-representing functions. In particular we show that there exist real numbers $\alpha \gt 1$ such that:

$$\lfloor \alpha^{2^n} \rfloor$$

is prime for all $n\in \Bbb N$. Indeed the set consisting of such numbers $\alpha$ has the cardinality of the continuum.

So basically is a refinement of Mills'constant that is, to my surprise, able to provide a prime representing function working in the quadratic intervals $[n^2,(n+1)^2]$ (if I did not understand wrongly, this is for big enough $n$).

My doubt regarding this is: as far as I can recall, Mills'constant is based on powers of three, $\lfloor A^{3^n} \rfloor$, and we still do not know if Legendre's conjecture is true or not, so there is not proof for a constant capable of making a prime-representing function like $\lfloor A^{2^n} \rfloor$. But according to the results of Matomäki, the study of the state of the art of prime gaps (and assuming that the Riemann hypothesis is true , as it is stated in the paper) makes possible to develop such power-of-two-based Mills-like constant.

So my question is: does the existence of such power of two Mills-like constant imply that Legendre's conjecture is true? where is the trick, or where is the point I misunderstood? maybe can the constant exist, but Legendre's conjecture can still be false without being a contradiction? Thank you!

Update 2017/06/06.

My perception is that Matomäki's proof shows that the existence of the constant implies that at least exist one infinite set of quadratic intervals $[N^2,(N+1)^2]$, indeed associated to the primes generated by the constant, each one of them containing at least one prime (indeed the one generated by the constant). So my guessing is that it is safe to say that (1) Still Legendre's conjecture is not proved to be true for all the existing quadratic intervals $[N^2,(N+1)^2]$, (2) Legendre's conjecture is true for those intervals associated to the primes generated by Matomäki's constant. A confirmation or not of this reasoning would be very appreciated.

Answer 1

I wrote Professor Matomäki (thank you very much for your time and kind response!), and she kindly gave me the following explanation that I just have formatted to write the answer (basically yes, Legendre's conjecture is true for those intervals):

The correspondence is the other way round, the paper uses the fact that Legendre's conjecture is true for most $N$, which is Lemma $7$ in the paper. That's the crucial ingredient which allows to take a power of $2$ in place of $3$ (or in place of $\frac{40}{19}$ following from Baker-Harman-Pintz on prime gaps).

Here is the aforementioned Lemma $7$:

Answer 2

The Legendre conjecture

for every $n$ there is a prime $ p \in [n^2,(n+1)^2]$

is the same as

for every $a_0 \ge 3$ there exists $\alpha \in [a_0,a_0+1]$ such that $a_n = \lfloor \alpha^{2^n} \rfloor$ is prime for every $n \ge 1$

This is because for every $n$ and for every $k \in [a_n^2,(a_n+1)^2]$, we can refine $\alpha$ without changing $a_0, \ldots, a_n$ to obtain $$a_{n+1} = \lfloor \alpha^{2^n} \rfloor = k$$