This article is an expanded version of my talk at the Gathering for Gardner, 2012.

I propose a class of numeral systems where numbers are represented by Dyck words, with the systems arising from a generalization of prime factorization. After describing two proper subsets of the Dyck language capable of uniquely representing all natural and rational numbers respectively, I consider "Dyck-complete" languages, in which every member of the Dyck language represents a number. I conclude by suggesting possible research directions.

In this paper, we reveal an internal structure within Dedekind numbers, demonstrating that they can be expressed as polynomials of powers of 2. This discovery is based on innovative concepts and methods, offering a new perspective on the nature of these numbers.

The Collatz variations pattern seems not to have any recurrence relation between numbers. But knowing that there is at least a natural number that converges after several iterations we construct a function $f_{X,Y}$ that is equal to the value of convergence for all convergent sequences. A canonical decomposition can be expressed for such numbers.

Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$. This has been generalized for any Positive Linear Recurrence Sequence (PLRS), which informally is a sequence satisfying a homogeneous linear recurrence with a positive leading coefficient and non-negative integer coefficients. These decompositions are generalizations of base $B$ decompositions. In...

Can we do arithmetic in a completely different way, with a radically different data structure? Could this approach provide practical benefits, like operations on giant numbers while having an average performance similar to traditional bitstring representations? While answering these questions positively, our tree based representation described in this paper comes with a few extra benefits: it compresses giant numbers such that, for instance, the largest known prime number a...

e use Prolog as a flexible meta-language to provide executable specifications of some fundamental mathematical objects and their transformations. In the process, isomorphisms are unraveled between natural numbers and combinatorial objects (rooted ordered trees representing hereditarily finite sequences and rooted ordered binary trees representing G\"odel's System {\bf T} types). This paper focuses on an application that can be seen as an unexpected "paradigm shift": we prov...

Algorithms can be used to prove and to discover new theorems. This paper shows how algorithmic skills in general, and the notion of invariance in particular, can be used to derive many results from Euclid's algorithm. We illustrate how to use the algorithm as a verification interface (i.e., how to verify theorems) and as a construction interface (i.e., how to investigate and derive new theorems). The theorems that we verify are well-known and most of them are included in st...

This paper is about computability. I claim the likely existence of a program DoesHalt(Program, Input) such that DoesHalt( HaltsOnItself, AntiSelf ) halts with resounding 'NO'. HaltsOnItself( Program ) is simply DoesHalt( Program, Program ). AntiSelf() is a self-referential self-contradictory program that loops when HaltsOnItself() returns 'YES' and halts when HaltsOnItself() returns 'NO'.

This paper presents mathematics as a general science of computation in a way different from the tradition. It is based on the radical philosophical standpoint according to which the content, meaning and justification of experience lies in its precise formulation. The requirement on precise, formal content discloses the relational structure of (mathematical) experience, and gives a new meaning to the `ideal' objects beyond concrete forms. The paper also provides a systematic r...