An arithmetic property of certain rational powers

Speaker: 

Gerry Myerson

Affiliation: 

Macquarie

Date: 

Wed, 24/07/2019 - 3:00pm

Venue: 

RC-4082, The Red Centre, UNSW

Abstract: 

In 1967, Forman and Shapiro proved

Theorem 1. The sequences $a_n = [(4/3)^n]$ and $a_n = [(3/2)^n]$ each contain infinitely many composites.

In 2005, Dubickas and Novikas proved these results:

Theorem 2. Let $\alpha > 0$ be a real number, and let $r$ be in the set $\{2, 3, 4, 5, 6, 3/2, 4/3, 5/4\}$. Then the sequence $a_n= [\alpha r^n]$ contains infinitely many primes.

Theorem 3. Let $\alpha > 0$ be a real number. Then the sequence $a_n = [\alpha(5/2)^n] - 1$ contains infinitely many composites.

Our main results are:

Theorem 4. For any non-zero $\alpha$, $\#\{n \le x : [r^n \alpha] {\rm\ is\ composite}\} \gg \log x$ for $r = 4/3$, $3/2$, $-3/2$, $-4/3$, $-5/4$, and $-6/5$.

Theorem 5. For any non-zero $\alpha$, $\#\{n \le x : [r^n\alpha - \beta_n] {\rm\ is\ composite}\} \gg \log x$ if

(a) $r=3/2$ and $7/5 < \beta_n < 8/5$ for all $n$, or
(b) $r=5/2$ and $13/21 < \beta_n < 22/21$ for all $n$.

Theorem 6. For all real $\beta$, and for all non-zero real $\alpha$, $[2^n\alpha - \beta]$ is composite infinitely often.

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