Related papers: A Complete Congruence System for the Erdos-Straus …
Using Eulerian and Euler numbers, we establish congruences concerning sums involving harmonic numbers, tangent numbers and Genocchi numbers.
A key example in Borger's theory of $\Lambda$-structure is toric $\Lambda$-structure. We prove a resolution of singularities result for embedded toric $\Lambda$-schemes by applying an algorithm of Bierstone and Milman for toric varieties…
The notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for $K_2$-algebras \cite{6}, is introduced. It is proved that the congruences of the principal MS-algebras $L$ correspond to the MS-congruence…
Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove an arithmetic analog of…
Reynolds' theory of relational parametricity formalizes parametric polymorphism for System F, thus capturing the idea that polymorphically typed System F programs always map related inputs to related results. This paper shows that Reynolds'…
The Casas-Alvero conjecture predicts that every univariate polynomial over an algebraically closed field of characteristic zero sharing a common factor with each of its Hasse-Schmidt derivatives is a power of a linear polynomial. The…
In this work, we determined the general terms of all almost balancing numbers of first and second type in terms of balancing numbers and conversely we determined the general terms of all balancing numbers in terms of all almost balancing…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
We adapt ideas of Phong, Stein and Sturm and ideas of Ikromov and M\"uller from the continuous setting to various discrete settings, obtaining sharp bounds for exponential sums and the number of solutions to polynomial congruences for…
In this paper we prove some new series for $1/\pi$ as well as related congruences. We also raise several new kinds of series for $1/\pi$ and present some related conjectural congruences involving representations of primes by binary…
A completeness conjecture is advanced concerning the free small-colimit completion P(A) of a (possibly large) category A. The conjecture is based on the existence of a small generating-cogenerating set of objects in A. We sketch how the…
In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible…
We survey some results that provide different versions of classical results through different summability methods. Specifically, in order to adapt such classical results, we analyze which properties should satisfy the summability methods.…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…
E prover is a state-of-the-art theorem prover for first-order logic with equality. E prover is built around a saturation loop, where new clauses are derived by inference rules from previously derived clauses. Selection of clauses for the…
We present an abstract unifying framework for interpreting Stone-type dualities; several known dualities are seen to be instances of just one topos-theoretic phenomenon, and new dualities are introduced. In fact, infinitely many new…
Recently the problem of constructing a perfect Euler cuboid was related with three conjectures asserting the irreducibility of some certain three polynomials depending on integer parameters. In this paper a partial result toward proving the…
The Erd\"os-S\'os conjecture states that if $G$ is a graph with average degree more than $k-1$, then G contains every tree of $k$ edges. A spider is a tree with at most one vertex of degree more than 2. In this paper, we prove that…
The Schinzel Hypothesis is a conjecture about irreducible polynomials in one variable over the integers: under some standard condition, they should assume infinitely many prime values at integers. We consider a relative version: if the…
In this article, we study word equations in free semigroups and the conjecture that the existence of infinitely many solutions entails the existence of solutions with arbitrarily large exponent of periodicity. We examine this question in…