M ar 2 00 7 Primes in a prescribed arithmetic progression dividing the sequence { a k + b k } ∞ k = 1
نویسنده
چکیده
Given positive integers a, b, c and d such that c and d are coprime we show that the primes p ≡ c(mod d) dividing ak + bk for some k ≥ 1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne. Mathematics Subject Classification (2001). 11N37, 11R45.
منابع مشابه
On rainbow 4-term arithmetic progressions
{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...
متن کاملOD-characterization of Almost Simple Groups Related to displaystyle D4(4)
Let $G$ be a finite group and $pi_{e}(G)$ be the set of orders of all elements in $G$. The set $pi_{e}(G)$ determines the prime graph (or Grunberg-Kegel graph) $Gamma(G)$ whose vertex set is $pi(G)$, the set of primes dividing the order of $G$, and two vertices $p$ and $q$ are adjacent if and only if $pqinpi_{e}(G)$. The degree $deg(p)$ of a vertex $pin pi(G)$, is the number of edges incident...
متن کاملThe power digraphs of safe primes
A power digraph, denoted by $G(n,k)$, is a directed graph with $Z_{n}={0,1,..., n-1}$ as the set of vertices and $L={(x,y):x^{k}equiv y~(bmod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper, the structure of $G(2q+1,k)$, where $q$ is a Sophie Germain prime is investigated. The primality tests for the integers of the form $n=2q+1$ are established in terms of th...
متن کاملSeven consecutive primes in arithmetic progression
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. In 1967, the first such sequence of 6 consecutive primes in arithmetic progression was found. Searching for 7 consecutive primes in arithmetic progression is difficult because it is necessary that a prescribed set of at least 1254 numbers between the first and last prime all be composi...
متن کاملDirichlet’s Theorem on Primes in Arithmetic Sequences Math 129 Final Paper
Dirichlet’s theorem on primes in arithmetic sequences states that in any arithmetic progression m,m + k, m + 2k, m + 3k, . . ., there are infinitely many primes, provided that (m, k) = 1. Euler first conjectured a result of this form, claiming that every arithmetic progression beginning with 1 contained an infinitude of primes. The theorem as stated was conjectured by Gauss, and proved by Diric...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007