Related papers: Some Tribonacci Conjectures
Following a recent paper of Anselmo et al., we consider $m \times n$ rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved with a finite automaton. We also generalize the result to…
The article provides a counterexample to a conjecture by Blocki-Zwonek.
Certain famous combinatorial sequences, such as the Catalan numbers and the Motzkin numbers, when taken modulo a prime power, can be computed by finite automata. Many theorems about such sequences can therefore be proved using Walnut, which…
We study the prefixes of the Fibonacci word that end with a cube. Using Walnut we obtain an exact description of the positions of the Fibonacci word at which a cube ends.
The recently developed proof of Fermat's Last Theorem is very lengthy and difficult, so much so as to be beyond all but a small body of specialists. While certainly of value in the developments that resulted, that proof could not be, nor…
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (arXiv:1907.00205), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued…
We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value…
We provide a method, based on automata theory, to mechanically prove the correctness of many numeration systems based on Fibonacci numbers. With it, long case-based and induction-based proofs of correctness can be replaced by simply…
This paper describes a method used to construct infinitely many probable counterexamples of the abc conjecture over the rational integers.
We show how the software Walnut can be used to obtain concise proofs of results concerning variants of the famous Wythoff game, in which blocking maneuvers or terminal positions are added, as discussed respectively by Larsson (2011) and…
The direct or algorithmic approach for the Jacobian problem, consisting of the direct construction of the inverse polynomials is proposed. The so called principle and derived Jacobi conditions are proposed and discussed. The algorithmic…
A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.
Probabilistic argumentation allows reasoning about argumentation problems in a way that is well-founded by probability theory. However, in practice, this approach can be severely limited by the fact that probabilities are defined by adding…
We study a class of infinite words $x_k$ , where $k$ is a positive integer, recently introduced by J. Shallit. This class includes the Thue-Morse sequence $x_1$, the Fibonacci-Thue-Morse sequence $x_2$, and the Allouche-Johnson sequence…
We introduce the $\omega$-Vaught's conjecture, a strengthening of the infinitary Vaught's conjecture. We believe that if one were to prove the infinitary Vaught's conjecture in a structural way without using techniques from higher recursion…
Motivated by the variations of Sarnak's conjecture due to El Abdalaoui, Kulaga-Przymus, Lemanczyk, De La Rue and by the observation that the Mobius function is a good weight (with limit zero) for the polynomial pointwise ergodic theorem in…
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
We state and prove a new closure theorem closely related to the classical closure theorems of Poncelet and Steiner. Along the way, we establish a number of theorems concerning conic sections.
We present two related conjectures, arising in work on i-matchings in random r-regular bipartite graphs. The conjectures themselves are easily stated and involve only basic properties of convergent power series. One formulation involves…