1,245 research outputs found
New Lower Bounds for van der Waerden Numbers Using Distributed Computing
This paper provides new lower bounds for van der Waerden numbers. The number
is defined to be the smallest integer for which any -coloring
of the integers admits monochromatic arithmetic progression of
length ; its existence is implied by van der Waerden's Theorem. We exhibit
-colorings of that do not contain monochromatic arithmetic
progressions of length to prove that . These colorings are
constructed using existing techniques. Rabung's method, given a prime and a
primitive root , applies a color given by the discrete logarithm base
mod and concatenates copies. We also used Herwig et al's
Cyclic Zipper Method, which doubles or quadruples the length of a coloring,
with the faster check of Rabung and Lotts. We were able to check larger primes
than previous results, employing around 2 teraflops of computing power for 12
months through distributed computing by over 500 volunteers. This allowed us to
check all primes through 950 million, compared to 10 million by Rabung and
Lotts. Our lower bounds appear to grow roughly exponentially in . Given that
these constructions produce tight lower bounds for known van der Waerden
numbers, this data suggests that exact van der Waerden Numbers grow
exponentially in with ratio asymptotically, which is a new conjecture,
according to Graham.Comment: 8 pages, 1 figure. This version reflects new results and reader
comment
Polignac Numbers, Conjectures of Erd\"os on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture
In the present work we prove a number of surprising results about gaps
between consecutive primes and arithmetic progressions in the sequence of
generalized twin primes which could not have been proven without the recent
fantastic achievement of Yitang Zhang about the existence of bounded gaps
between consecutive primes. Most of these results would have belonged to the
category of science fiction a decade ago. However, the presented results are
far from being immediate consequences of Zhang's famous theorem: they require
various new ideas, other important properties of the applied sieve function and
a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, and
Zhang, respectively
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
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