1,283 research outputs found
Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2017) and another new proof
In this article, we provide a comprehensive historical survey of 183
different proofs of famous Euclid's theorem on the infinitude of prime numbers.
The author is trying to collect almost all the known proofs on infinitude of
primes, including some proofs that can be easily obtained as consequences of
some known problems or divisibility properties. Furthermore, here are listed
numerous elementary proofs of the infinitude of primes in different arithmetic
progressions.
All the references concerning the proofs of Euclid's theorem that use similar
methods and ideas are exposed subsequently. Namely, presented proofs are
divided into 8 subsections of Section 2 in dependence of the methods that are
used in them. {\bf Related new 14 proofs (2012-2017) are given in the last
subsection of Section 2.} In the next section, we survey mainly elementary
proofs of the infinitude of primes in different arithmetic progressions.
Presented proofs are special cases of Dirichlet's theorem. In Section 4, we
give a new simple "Euclidean's proof" of the infinitude of primes.Comment: 70 pages. In this extended third version of the article, 14 new
proofs of the infnitude of primes are added (2012-2017
Quantitative bounds in the polynomial Szemer\'edi theorem: the homogeneous case
We obtain quantitative bounds in the polynomial Szemer\'edi theorem of
Bergelson and Leibman, provided the polynomials are homogeneous and of the same
degree. Such configurations include arithmetic progressions with common
difference equal to a perfect kth power.Comment: v2. Title changed and substantial alterations to exposition. v3.
Referee comments incorporated. v4 Formatted using Discrete analysis style
fil
-adic quotient sets
For , the question of when is dense in the positive real numbers has been examined by
many authors over the years. In contrast, the -adic setting is largely
unexplored. We investigate conditions under which is dense in the
-adic numbers. Techniques from elementary, algebraic, and analytic number
theory are employed in this endeavor. We also pose many open questions that
should be of general interest.Comment: 24 page
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