One of the most fundamental questions in Biology or Artificial Intelligence is how the human brain performs mathematical functions. How does a neural architecture that may organise itself mostly through statistics, know what to do? One possibility is to extract the problem to something more abstract. This becomes clear when thinking about how the brain handles large numbers, for example to the power of something, when simply summing to an answer is not feasible. In this paper...

We introduce and study the recursive divisor function, a recursive analog of the usual divisor function: $\kappa_x(n) = n^x + \sum_{d\lfloor n} \kappa_x(d)$, where the sum is over the proper divisors of $n$. We give a geometrical interpretation of $\kappa_x(n)$, which we use to derive a relation between $\kappa_x(n)$ and $\kappa_0(n)$. For $x \geq 2$, we observe that $\kappa_x(n)/n^x < 1/(2-\zeta(x))$. We show that, for $n \geq 2$, $\kappa_0(n)$ is twice the number of ordered...

Basic arithmetic is the cornerstone of mathematics and computer sciences. In arithmetic, 'division by zero' is an undefined operation and any attempt at extending logic for algebraic division to incorporate division by zero has resulted in paradoxes and fallacies. However, there is no proven theorem or mathematical logic that suggests that, defining logic for division by zero would result in break-down of theory. Basing on this motivation, in this paper, we attempt at logical...

The present paper shows meta-programming turn programming, which is rich enough to express arbitrary arithmetic computations. We demonstrate a type system that implements Peano arithmetics, slightly generalized to negative numbers. Certain types in this system denote numerals. Arithmetic operations on such types-numerals - addition, subtraction, and even division - are expressed as type reduction rules executed by a compiler. A remarkable trait is that division by zero become...

In this paper Euler shows how, if we have recursive functions f,g,h and an infinite sequence A,B,C,... which satisfies fA=gB+hC, f'B=g'C+h'D, f''C=g''D+h''E, f'''D=g'''E+h'''F, etc., where the primes denote an index not a derivative, then we can find a continued fraction for fA/B.

This paper contains an algebraic constructive and self-contained account of the invariance rule of the digital root under division for an arbitrary natural basis representation. Both the cases of repeating and non-repeating fractionals are treated. In the preliminary section some known results such as the uniqueness in the representation of a fraction are discussed for both the finite and infinite bases cases. Simple examples are introduced throughout the text for illustrativ...

This paper grew out of the observation that the possibilities of proof by induction and definition by recursion are often confused. The paper reviews the distinctions. The von Neumann construction of the ordinal numbers includes a construction of natural numbers as a special kind of ordinal. In any case, the natural numbers can be understood as composing a free algebra in a certain signature, {0,s}. The paper here culminates in a construction of, for each algebraic signature ...

Write A<=B if there is an injection from A to B, and A==B if there is a bijection. We give a simple proof that for finite n, nA<=nB implies A<=B. From the Cantor-Bernstein theorem it then follows that nA==nB implies A==B. These results have a long and tangled history, of which this paper is meant to be the culmination.

Literature considers under the name \emph{unimaginable numbers} any positive integer going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathematical definition. This simply means that research in this topic must always consider shortened representations, usually involving \emph{recursion}, to even being able to describe such numbers.\par\medskip One of the most known methodologies to conceive s...

After a short review of the Method of Recursive Counting we introduce a general algebraic description of recursive lattice building. This provides a rigorous framework for discussion of method's limitations.