Arithmetic Progressions, Prime Numbers, and Squarefree Integers
نویسندگان
چکیده
منابع مشابه
Prime Numbers in Certain Arithmetic Progressions
We discuss to what extent Euclid’s elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet’s theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod k ) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.
متن کاملGaps between Prime Numbers and Primes in Arithmetic Progressions
The equivalence of the two formulations is clear by the pigeon-hole principle. The first one is psychologically more spectacular: it emphasizes the fact that for the first time in history, one has proved an unconditional existence result for infinitely many primes p and q constrained by a binary condition q − p = h. Remarkably, this already extraordinary result was improved in spectacular fashi...
متن کاملSquarefree Smooth Numbers and Euclidean Prime Generators
We show that for each prime p > 7, every residue mod p can be represented by a squarefree number with largest prime factor at most p. We give two applications to recursive prime generators akin to the one Euclid used to prove the infinitude of primes.
متن کاملSquarefree polynomials and Möbius valuesin short intervalsand arithmetic progressions
Abstract. We calculate the mean and variance of sums of the Möbius function μ and the indicator function of the squarefrees μ, in both short intervals and arithmetic progressions, in the context of the ring Fq[t] of polynomials over a finite field Fq of q elements, in the limit q → ∞. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent eq...
متن کاملPalindromic Numbers in Arithmetic Progressions
Integers have many interesting properties. In this paper it will be shown that, for an arbitrary nonconstant arithmetic progression {an}TM=l of positive integers (denoted by N), either {an}TM=l contains infinitely many palindromic numbers or else 10|aw for every n GN. (This result is a generalization of the theorem concerning the existence of palindromic multiples, cf. [2].) More generally, for...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Czechoslovak Mathematical Journal
سال: 2004
ISSN: 0011-4642,1572-9141
DOI: 10.1007/s10587-004-6440-6