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On coincidences among quadratic fields generated by the Shanks sequence

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Speaker: 
Igor Shparlinski
Affiliation: 
UNSW
Date: 
Thu, 07/04/2016 - 14:00 - 14:50
Location: 
MPIM Lecture Hall
Parent event: 
Number theory downside up
Motivated by work of Shanks (1969), we study the distribution of the fields ${\mathbb Q}\big(\sqrt{f(g^n)}\big)$ for a polynomial $f \in {\mathbb Z}[X]$ and an integer $g >1$. Using a variety of known results and techniques such as the abundance of shifted primes with a large prime divisor (R. Baker and G. Harman) and the square sieve (R. Heath-Brown) together with new bounds on character sums, we improve an upper bound of Luca and Shparlinski (2009) on the number of $n \in \{M+1,\ldots,M+N\}$ with ${\mathbb Q}\bigl(\sqrt{f(g^n)}\bigr) ={\mathbb Q}\bigl(\sqrt{s}\bigr)$ for a given squarefree integer $s$, individually and on average over $s \le S$. (Joint work with Bill Banks.)
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