Fifteen Eighty Four

The basso continuo of these essays is Euclid’s algorithm. The author wants readers to discover that almost every page contains the algorithm either visibly or implicitly or in disguised forms. Readers should eventually be amazed that the algorithm is so simple yet deep and strong. Moving to the study of higher algebraic structures, readers will perceive that the same will continue. I am relating here that my own awakening to this marvel in mathematics all began with a somewhat untypical experience:

My mother carried me on her back as we fled a port city and evacuated to a faraway village upstream on a major river. A few years later, I entered elementary school there. My first report card read, “Yoichi has some difficulty understanding numbers.” My mother didn’t seem to mind it at all but let me play as much as I like. As the snow melted, I rolled around with my dog in the spring rice paddies, where the vetch bloomed like gorgeous carpets. I collected mulberries and made jam. I chased loaches in a stream, being once frightened by an ominous red-bellied snake. Insects were also of great interest. Magnificent stag-beetles gathered around the sap of oak trees, and they were apparently more numerous early in the morning. That was a gem discovery I never revealed to anyone. I quietly slipped out of my bed in dawn and ran to the target tree. I packed a few catches in a bamboo basket, returned home, and went to sleep again ¾  I am doing essentially the same in my mathematical research.

With memories of those days in mind, I returned to the port city as a third-grader in elementary school. I did not like my new school life very much as I came from a deeply rural village and needed considerable efforts to catch up with my classmates. My mother again let me do as I like. I continued my daily life of wonderful discovery; so for instance I was mesmerized by mysteriously beautiful butterflies (papilio maackii) fluttering about in the dimly sun-spotted wasabi fields.

In summer vacation, I made daily visits to the quay with my dog to fish for horse mackerel. Masts appeared on the horizon beyond which is the sacred Mt. Fuji, and huge ocean-going ships reveal themselves. I began to yearn to see exotic foreign countries. I started visiting the local public library. It was wonderful for me to be able to sit on the wooden floor with my shoes on and browse illustrated books freely. I acquired exciting knowledge such as “fish raining in a desert” and “photographs of nebulae from hundreds of millions of years ago (in fact, hundreds of millions of light-years away)” and enthusiastically tried to share them with my mother and neighbores. I would later learn my mother’s feelings at the time: “I wonder what will become of my son.”

Then I somehow learned about terms like “factorization” and “greatest common divisor”; or rather, I came across something similar. Something else began to emerge, replacing the streams and forests, and I became captivated by the countless integers that lay there. I became engrossed in division and common divisors. And a great leap in my mind occurred. From somewhere, I learnt Euclid’s algorithm, purely as a device of course. It appeared to me like an all-purpose knife. The excitement remains the same to this day. Since that experience, my feelings toward number theory have not changed at all. While I still have difficulty understanding numbers (!), the joy of immersing myself in the mysteries of integers has just deepened.

Title: Essays in Classical Number Theory

ISBN: 9781009504553

Author: Yoichi Motohashi