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1. CMB 2012 (vol 57 pp. 105)
On the Counting Function of Elliptic Carmichael Numbers We give an upper bound for the number elliptic Carmichael numbers $n \le x$
that have recently been introduced by J. H. Silverman in the case of an elliptic curve without complex multiplication (non CM). We also discuss
several possible ways for further improvements.
Keywords:elliptic Carmichael numbers, applications of sieve methods Categories:11Y11, 11N36 |
2. CMB 2008 (vol 51 pp. 399)
Linear Equations with Small Prime and Almost Prime Solutions Let $b_1, b_2$ be any integers such that
$\gcd(b_1, b_2)=1$ and $c_1|b_1|<|b_2|\leq c_2|b_1|$, where
$c_1, c_2$ are any given positive constants. Let $n$ be any
integer satisfying $\{gcd(n, b_i)=1$, $i=1,2$. Let $P_k$ denote
any integer with no more than $k$ prime factors, counted according
to multiplicity. In this paper, for almost all $b_2$, we prove (i)
a sharp lower bound for $n$ such that the equation $b_1p+b_2m=n$
is solvable in prime $p$ and almost prime $m=P_k$, $k\geq 3$
whenever both $b_i$ are positive, and (ii) a sharp upper bound for the
least solutions $p, m$ of the above equation whenever $b_i$ are
not of the same sign, where $p$ is a prime and $m=P_k, k\geq 3$.
Keywords:sieve method, additive problem Categories:11P32, 11N36 |
3. CMB 2005 (vol 48 pp. 16)
On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves Let $ E $ be an elliptic curve defined over
$\Q,$ of conductor $N$ and without complex multiplication. For any
positive integer $l$, let $\phi_l$ be the Galois representation
associated to the $l$-division points of~$E$. From a celebrated
1972 result of Serre we know that $\phi_l$ is surjective for any
sufficiently large prime $l$. In this paper we find conditional
and unconditional upper bounds in terms of $N$ for the primes $l$
for which $\phi_l$ is {\emph{not}} surjective.
Categories:11G05, 11N36, 11R45 |