The method used by Cantor to demonstrate the uncountability of the real numbers is applied to a proof showing that the set of natural numbers is uncountable; the error in the argument is discussed. (DT)

Some procedures are developed for testing divisibility by prime numbers composed of two or more digits. Accelerating the tests is also considered. (Contains 2 tables.)

It is one thing to be able to count and share items proficiently, but it is another thing to know how counting and sharing establish and identify quantity. The aim of the study was to identify which measures of numerical knowledge predict children's success on simple number problems, where counting and set equivalence are at issue. Seventy-two…

Patients with corticobasal degeneration (CBD) have calculation impairments. This study examined whether impaired number knowledge depends on verbal mediation. We focused particularly on knowledge of very small numbers, where there is a precise relationship between a cardinality and its number concept, but little hypothesized role for verbal…

In this article, the author relates his unhappy experience in learning about prime numbers at secondary school. To introduce primes, a teacher first told students a definition of a prime number, then students were taught how to find prime numbers. Students defined and listed them and at some later point were tested on their memory of both the…

Twenty-three kindergarten and first grade children were asked to articulate the meaning and the need for punctuation marks in a list of numerals showing prices for a list of items. Despite not having been schooled yet formally on the use and roles of numerical punctuation, many children gave similar explanations regarding the purpose of…

A general method is presented for evaluating the sums of "m"th powers of the integers that can, and that cannot, be represented in the two-element Frobenius problem. Generating functions are introduced and used for that purpose. Explicit formulas for the desired sums are obtained and specific examples are discussed.

Does speaking a language without number words change the way speakers of that language perceive exact quantities? The Piraha are an Amazonian tribe who have been previously studied for their limited numerical system [Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. "Science 306", 496-499]. We show that the Piraha have…

This paper presents the early investigations about the nature of sound of the Pythagoreans, and how they started a tradition that remains valid up to present times--the use of numbers in representing natural reality. It will touch on the Pythagorean notion of musical harmony, which was extended to the notion of universal harmony. How the…

Two identities for the Bernoulli and for the Euler numbers are derived. These identities involve two special cases of central combinatorial numbers. The approach is based on a set of differential identities for the powers of the secant. Generalizations of the Mittag-Leffler series for the secant are introduced and used to obtain closed-form…

Based on an empirical study, we explore children's primary and secondary perceptions on infinity. When discussing infinity, children seem to highlight three categories of primary perceptions: processional, topological, and spiritual. Based on their processional perception, children see the set of natural numbers as being infinite and endow Q with…

This article explores prospective teachers' understandings of one million to gain insights into the development of adult understanding of large numbers. Themes in the prospective teachers' work included number associated with a quantity of objects, number as an abstraction, and additive and multiplicative approaches. The authors suggest that the…

The principle of the permanence of the rules of calculation is contrasted with the concretization permanence principle. Both apply to situations where some arithmetical operation known to children for numbers of a certain kind is to be extended to include further numbers. (MNS)