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On some formulae related to Euler sums
Using the Ramanujan summation method, we derive some unusual formulas for a class of Euler sums (including divergent Euler sums) similar to the...
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On the Critical Exponent for k-Primitive Sets
A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum ∑1/( n log n ) for n ...
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The equal–sum–free subset problem
Given a set W of positive integers, a set I ⊆ W is independent if all the partial sums in I are distinct. We prove estimates on the maximum size of...
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Polynomial Schur’s Theorem
We resolve the Ramsey problem for { x,y,z : x + y = p ( z )} for all polynomials p over ℤ. In particular, we characterise all polynomials that are...
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Congruence classes and maximal nonbases
The set A is an asymptotic nonbasis of order h for an additive abelian semigroup X if there are infinitely many elements of X not in the h -fold...
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Some Identities of Cauchy Numbers Associated with Continued Fractions
In this paper, the n -th convergent of the generating function of Cauchy numbers is explicitly given. As an application, we give some new identities...
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Subsets in Linear Spaces over the Finite Field F3 Uniquely Determined by Their Pairwise Sums Collection
Let F 3 n be an n -dimensional linear space over the finite field F 3 . Let A = { a 1 , a 2 ,..., a N } be a set in F 3 n and A + A be the collection of sums of...
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Geometric progressions in syndetic sets
In order to investigate multiplicative structures in additively large sets, Beiglböck et al. raised a significant open question as to whether or not...
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Divisibility problems for function fields
We investigate three combinatorial problems considered by Erdős, Rivat, Sárközy and Schön regarding divisibility properties of sum sets and sets of...
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Partitions of the set of natural numbers and symplectic homology
We illustrate a somewhat unexpected relation between symplectic geometry and combinatorial number theory by proving Tamura’s theorem on partitions of...
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On symmetries of roots of rational functions and the classification of rational function solutions of functional equations arising from multiplication of quantum integers with prime semigroup supports
The study of quantum integers and their operations is closely related to the studies of symmetries of roots of polynomials and of fundamental...