Results 1 to 10 of about 272 (33)
New lower bounds for van der Waerden numbers
Forum of Mathematics, Pi, 2022 We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$.
Ben Green
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High-entropy dual functions over finite fields and locally decodable codes
Forum of Mathematics, Sigma, 2021 We show that for infinitely many primes p there exist dual functions of order k over ${\mathbb{F}}_p^n$ that cannot be approximated in $L_\infty $-distance by polynomial phase functions of degree $k-1$. This answers in the negative a natural finite-field
Jop Briët, Farrokh Labib
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Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
Open Mathematics, 2023 We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer k≥2k\ge 2, there are ≫logx\gg \hspace{0.25em}\log x Lucas non-Wieferich primes p≤xp\le x such that p≡±1(modk)p\equiv ...
Anitha K. +2 more
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A characterization of covering equivalence [PDF]
, 2007 Let A={a_s(mod n_s)}_{s=1}^k and B={b_t(mod m_t)}_{t=1}^l be two systems of residue classes. If |{1\le s\le k: x=a_s (mod n_s)}| and |{1\le t\le l: x=b_t (mod m_t)}| are equal for all integers x, then A and B are said to be covering equivalent.
Pan, Hao, Sun, Zhi-Wei
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Covering an arithmetic progression with geometric progressions and vice versa [PDF]
, 2013 We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression A of real numbers and a sufficiently large integer n (depending on A), there needs at least Cn geometric progressions to cover the ...
Sanna, Carlo
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ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS
Forum of Mathematics, Sigma, 2016 We show that if $A\subset \{1,\ldots ,N\}$ does not contain any nontrivial solutions to the equation
TOMASZ SCHOEN, OLOF SISASK
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The elementary symmetric functions of a reciprocal polynomial sequence [PDF]
, 2014 Erd\"{o}s and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers. Recently, Wang and
Hong, Shaofang +3 more
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Goldbach Conjecture and the least prime number in an arithmetic progression [PDF]
, 2010 In this Note, we try to study the relations between the Goldbach Conjecture and the least prime number in an arithmetic progression. We give a new weakened form of the Goldbach Conjecture.
Zhang, Shaohua
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On covers of abelian groups by cosets [PDF]
, 2008 Let G be any abelian group and {a_sG_s}_{s=1}^k be a finite system of cosets of subgroups G_1,...,G_k. We show that if {a_sG_s}_{s=1}^k covers all the elements of G at least m times with the coset a_tG_t irredundant then [G:G_t]\le 2^{k-m} and ...
Lettl, Günter, Sun, Zhi-Wei
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A sharp result on m-covers [PDF]
, 2005 Let A={a_s+n_sZ}_{s=1}^k be a finite system of arithmetic sequences which forms an m-cover of Z (i.e., every integer belongs at least to m members of A). In this paper we show the following sharp result: For any positive integers m_1,...,m_k and theta in
Pan, Hao, Sun, Zhi-Wei
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