On the infinitude of the prime numbers

Euclid’s elegant proof that there are infinitely many prime numbers is well known. Elder proved the same result, in fact a stronger one, byanalytical methods. This article gives an exposition of Euler’s proof introducing the necessary concepts along the way.     

Marin Mersenne, 1588–1648

Marin Mersenne was many things: scientist, mathematician and writer. He did original work on acoustics and on prime numbers; today his name is linked to a family of primes called ‘Mersenne primes’. But his greatest contribution was his work in propagating a culture of scientific inquiry in Europe — a culture which emphasized communication and dissemination and learning from one’s peers. The work done on the cycloid illustrates this theme beautifully. This article looks at some work for which Mersenne is best known.     

Combinatorial proofs and algebraic proofs — II

In Part I of this article we considered some binomial identities and also some identities involving the Fibonacci numbers, and proved them using methods which we described as ‘largely’ combinatorial. Now we shift our focus to number theory and to prime numbers in particular, and showcase some proofs having a strong combinatorial element. Throughout this article, p denotes an odd prime.     

A pizza saga Five delicacies of modern geometry

In this article we introduce the reader to some pretty results concerning a circle and regions within it defined by various line segments. We have framed the results in terms of a pizza, so we could well call this an introduction to Pizza Geometry! The novel aspect is the variety of approaches used to prove these results.     

Movement pattern and parameter learning in children: effects of feedback frequency

Reduced feedback during practice has been shown to be detrimental to movement accuracy in children but not in young adults. We hypothesized that the reduced accuracy is attributable to reduced movement parameter learning but not pattern learning in children. A rapid arm movement task that required the acquisition of a motorpattern scaled to specific spatial and temporal parameters was used to investigate the effects of feedback (FB) frequency (100% vs. 62% faded) on motor learning differences between 19 school-age children and 19 young adults. Adults and children practiced the task for 200 trials under the 100% or faded FB condition on day 1 and returned on day 2 for a no-FB retention test. On the retention test, children who practiced with reduced feedback performed with greater temporal parameter errors, but not pattern error than children who received frequent feedback. Motor skill learning in children is influenced byfeedback frequency during practice that affects parameter learning but not pattern learning.