Search results

Search: MSC category 11N32 ( Primes represented by polynomials; other multiplicative structure of polynomial values )

 Expand all        Collapse all Results 1 - 3 of 3

1. CMB 2012 (vol 56 pp. 844)

Shparlinski, Igor E.
 On the Average Number of Square-Free Values of Polynomials We obtain an asymptotic formula for the number of square-free integers in $N$ consecutive values of polynomials on average over integral polynomials of degree at most $k$ and of height at most $H$, where $H \ge N^{k-1+\varepsilon}$ for some fixed $\varepsilon\gt 0$. Individual results of this kind for polynomials of degree $k \gt 3$, due to A. Granville (1998), are only known under the $ABC$-conjecture. Keywords:polynomials, square-free numbersCategory:11N32

2. CMB 2007 (vol 50 pp. 409)

Luca, Florian; Shparlinski, Igor E.
 Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields We show that, for most of the elliptic curves $\E$ over a prime finite field $\F_p$ of $p$ elements, the discriminant $D(\E)$ of the quadratic number field containing the endomorphism ring of $\E$ over $\F_p$ is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over $\F_p$. Categories:11G20, 11N32, 11R11

3. CMB 2003 (vol 46 pp. 71)

Cutter, Pamela; Granville, Andrew; Tucker, Thomas J.
 The Number of Fields Generated by the Square Root of Values of a Given Polynomial The $abc$-conjecture is applied to various questions involving the number of distinct fields $\mathbb{Q} \bigl( \sqrt{f(n)} \bigr)$, as we vary over integers $n$. Categories:11N32, 11D41