46,385 research outputs found
Long zero-free sequences in finite cyclic groups
A sequence in an additively written abelian group is called zero-free if each
of its nonempty subsequences has sum different from the zero element of the
group. The article determines the structure of the zero-free sequences with
lengths greater than in the additive group \Zn/ of integers modulo .
The main result states that for each zero-free sequence of
length in \Zn/ there is an integer coprime to such that if
denotes the least positive integer in the congruence class
(modulo ), then . The answers to a number of
frequently asked zero-sum questions for cyclic groups follow as immediate
consequences. Among other applications, best possible lower bounds are
established for the maximum multiplicity of a term in a zero-free sequence with
length greater than , as well as for the maximum multiplicity of a
generator. The approach is combinatorial and does not appeal to previously
known nontrivial facts.Comment: 13 page
On the joint distribution of digital sums
AbstractLet s(n) be the sum of the digits of n written to the base b. We determine the joint distribution (modulo m) of the sequences s(k1n), β¦, s(kln). In the case where m and b β 1 are relatively prime, we find that their values are equally distributed among l-tuples of residue classes (modulo m)
Arithmetic-Progression-Weighted Subsequence Sums
Let be an abelian group, let be a sequence of terms
not all contained in a coset of a proper subgroup of
, and let be a sequence of consecutive integers. Let
which is a particular kind of weighted restricted sumset. We show that , that if , and also
characterize all sequences of length with . This
result then allows us to characterize when a linear equation
where are
given, has a solution modulo with all
distinct modulo . As a second simple corollary, we also show that there are
maximal length minimal zero-sum sequences over a rank 2 finite abelian group
(where and ) having
distinct terms, for any . Indeed, apart from
a few simple restrictions, any pattern of multiplicities is realizable for such
a maximal length minimal zero-sum sequence
Long -zero-free sequences in finite cyclic groups
A sequence in the additive group of integers modulo is
called -zero-free if it does not contain subsequences with length and
sum zero. The article characterizes the -zero-free sequences in of length greater than . The structure of these sequences is
completely determined, which generalizes a number of previously known facts.
The characterization cannot be extended in the same form to shorter sequence
lengths. Consequences of the main result are best possible lower bounds for the
maximum multiplicity of a term in an -zero-free sequence of any given length
greater than in , and also for the combined
multiplicity of the two most repeated terms. Yet another application is finding
the values in a certain range of a function related to the classic theorem of
Erd\H{o}s, Ginzburg and Ziv.Comment: 11 page
Motzkin numbers and related sequences modulo powers of 2
We show that the generating function for Motzkin
numbers , when coefficients are reduced modulo a given power of , can
be expressed as a polynomial in the basic series with coefficients being Laurent polynomials in and
. We use this result to determine modulo in terms of the binary
digits of~, thus improving, respectively complementing earlier results by
Eu, Liu and Yeh [Europ. J. Combin. 29 (2008), 1449-1466] and by Rowland and
Yassawi [J. Th\'eorie Nombres Bordeaux 27 (2015), 245-288]. Analogous results
are also shown to hold for related combinatorial sequences, namely for the
Motzkin prefix numbers, Riordan numbers, central trinomial coefficients, and
for the sequence of hex tree numbers.Comment: 28 pages, AmS-LaTeX; minor typos correcte
Category equivalences involving graded modules over path algebras of quivers
Let kQ be the path algebra of a quiver Q with its standard grading. We show
that the category of graded kQ-modules modulo those that are the sum of their
finite dimensional submodules, QGr(kQ), is equivalent to several other
categories: the graded modules over a suitable Leavitt path algebra, the
modules over a certain direct limit of finite dimensional multi-matrix
algebras, QGr(kQ') where Q' is the quiver whose incidence matrix is the n^{th}
power of that for Q, and others. A relation with a suitable Cuntz-Krieger
algebra is established. All short exact sequences in the full subcategory of
finitely presented objects in QGr(kQ), split so that subcategory can be given
the structure of a triangulated category with suspension functor the Serre
degree twist (-1); it is shown that this triangulated category is equivalent to
the "singularity category" for the radical square zero algebra kQ/kQ_{\ge 2}.Comment: Several changes made as a result of the referee's report. Added Lemma
3.5 and Prop. 3.6 showing that O is a generato
Classification theorems for sumsets modulo a prime
Let be the finite field of prime order and be a subsequence
of . We prove several classification results about the following
questions: (1) When can one represent zero as a sum of some elements of ?
(2) When can one represent every element of as a sum of some elements
of ? (3) When can one represent every element of as a sum of
elements of ?Comment: 35 pages, to appear in JCT
Every sufficiently large even number is the sum of two primes
The binary Goldbach conjecture asserts that every even integer greater than
is the sum of two primes. In this paper, we prove that there exists an
integer such that every even integer can be expressed as
the sum of two primes, where is the th prime number and . To prove this statement, we begin by introducing a type of double
sieve of Eratosthenes as follows. Given a positive even integer , we
sift from all those elements that are congruents to modulo or
congruents to modulo , where is a prime less than .
Therefore, any integer in the interval that remains unsifted is
a prime for which either or is also a prime. Then, we
introduce a new way of formulating a sieve, which we call the sequence of
-tuples of remainders. By means of this tool, we prove that there exists an
integer such that is a lower bound for the sifting
function of this sieve, for every even number that satisfies , where , which implies that can be expressed as the sum of two primes.Comment: 32 pages. The manuscript was edited for proper English language by
one editor at American Journal Experts (Certificate Verification Key:
C0C3-5251-4504-E14D-BE84). However, afterwards some changes have been made in
sections 1, 6, 7 and
- β¦