Related papers: Some Tribonacci Conjectures
Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and…
We present a lecture note on Thouvenot's proof of the Roth-Furstenberg theorem and joining proofs of Furstenberg's theorems on multiple progression average mixing for weakly mixing transformations.
The purpose of this note is to give an affirmative answer to a conjecture appearing in [Integral Transforms Spec. Funct. 26 (2015) 90-95].
We present a new proof of the Joints Theorem without taking derivatives. Then we generalize the proof to prove the Multijoints Conjecture and Carbery's generalization. All results are in any dimension over an arbitrary field.
We describe Greenberg's pseudo-null conjecture, and prove a result describing conditions under which the pseudo-null conjecture for a number field $K$ implies the conjecture for finite extensions of $K$. We then apply the result to the…
Jakob Bernoulli, working in the late 17th century, identified a gap in contemporary probability theory. He cautioned that it was inadequate to specify force of proof (probability of provability) for some kinds of uncertain arguments. After…
By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial…
We discuss the use of negative bases in automatic sequences. Recently the theorem-prover Walnut has been extended to allow the use of base (-k) to express variables, thus permitting quantification over Z instead of N. This enables us to…
We present a tool that automates termination proofs for recursive definitions by mining existing termination theorems.
In this note, infinite series involving Fibonacci and Lucas numbers are derived by employing formulae similar to that which Roger Ap\'ery utilized in his seminal paper proving the irrationality of $\zeta(3)$.
In this study, the new algebraic properties related to bivariate Fibonacci polynomials has been given. We present the partial derivatives of these polynomials in the form of convolution of bivariate Fibonacci polynomials. Also, we define a…
The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…
In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with…
We prove a conjecture due to Y. Last on Jacobi matrices.
In this paper, we survey some recent results on the Artin conjecture and discuss some aspects for the Artin conjecture.
In this article we provide further evidence for the monstrous proposal of Daniel Allcock, by giving a plausible but still conjectural explanation for the deflation relation in the Coxeter group quotient of the orbifold fundamental group.
Several recent results bring into focus the superintuitionistic nature of most notions of proof-theoretic validity, but little work has been done evaluating the consequences of these results. Proof-theoretic validity claims to offer a…
The Collatz Conjecture (also known as the 3x+1 Problem) proposes that the following algorithm will, after a certain number of iterations, always yield the number 1: given a natural number, multiply by three and add one if the number is odd,…
A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of…
Let $f$ be a transcendental entire function. By a result of Rippon and Stallard, there exist points whose orbit escapes arbitrarily slowly. By using a range of techniques to prove new covering results, we extend their theorem to prove the…