Numbers, Numbers, Numbers: From Nicomachus to Ramanujan
J.J. Tattersall
Two manuscripts, the Introduction to Arithmetic, by Nicomachus of Gerasa and Mathematics Useful for Understanding Plato by Theon of Smyrna were written in the second century A.D. They were the main sources of knowledge of formal Greek arithmetic in the Middle Ages. The books are philosophical in nature, contain few original results and no formal proofs. They abound, however, in intriguing number theoretic observations. We extend some of the results found in these ancient works and introduce several types of numbers that lend themselves naturally to undergraduate research, in particular, happy, sad, polite, Demlo, Niven, decimal columbian, and highly composite numbers.
Jim Tattersall received his undergraduate degree in mathematics from the University of Virginia, a Master’s degree in mathematics from the University of Massachusetts, and a Ph.D. degree in mathematics from the University of Oklahoma. On a number of occasions he has been a visiting scholar at the Department of Pure Mathematics and Mathematical Statistics at Cambridge University and a visiting fellow at Wolfson College, Cambridge. He spent the summer of 1991 as a visiting mathematician at the American Mathematical Society. In 1995‑1996, he spent eighteen months as a visiting professor at the U.S. Military Academy at West Point. He was given awards for distinguished service (1992) and distinguished college teaching (1997) from the Northeastern Section of the Mathematical Association of America (MAA). He is former President of the Canadian Society for History and Philosophy of Mathematics, and Associate Secretary of the MAA.
