ﻻ يوجد ملخص باللغة العربية
We look at upper bounds for the count of certain primes related to the Fermat numbers $F_n=2^{2^n}+1$ called elite primes. We first note an oversight in a result of Krizek, Luca and Somer and give the corrected, slightly weaker upper bound. We then assume the Generalized Riemann Hypothesis for Dirichlet L functions and obtain a stronger conditional upper bound.
Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $mathbb{F}_q$. A general result of our study is that $(a,b)leq
We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 geq p^theta$ for $theta=1/2+1/2000.$ This improves the work of Matomaki (2009) who obtained the result for $theta=1/2-varepsilon$ (with the added constrai
For an elliptic curve E/Q without complex multiplication we study the distribution of Atkin and Elkies primes l, on average, over all good reductions of E modulo primes p. We show that, under the Generalised Riemann Hypothesis, for almost all primes
We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varie
When the sequences of squares of primes is coloured with $K$ colours, where $K geq 1$ is an integer, let $s(K)$ be the smallest integer such that each sufficiently large integer can be written as a sum of no more than $s(K)$ squares of primes, all of