Amicable numbers

Amicable Numbers by René Oosterlinck.

On Monday, March 25, 2019 Yorika Kawaguchi, Mika Igarashi and Kazuki Kita, three PhD students came to Paris, at the European Space Agency, to present and discuss their work.

Yorika’s research is about the classical music and nationalism. She focuses her research on the     ideological perspective. During our meeting Yorika wanted to discuss about 19s century ‘s classical music and the politics in France. I therefore invited Nathalie Meusy whose family has been involved in setting up the Besançon International Music Festival. This festival is one of the oldest festivals of classical music, created in 1948, that takes place in the city of Besançon.

In 1959, Seiji Ozawa aged 24, won the first prize of the Conducting Competition of the Festival and came back to Besançon in the sixties and started a phenomenal career all over the world.

Nathalie and Yorika had an interesting discussion on Yorika’s research.

Mika Igarashi research is focused on human rights /diversity in Africa. Having spent several years in Africa myself mainly in Congo (ex – Zaïre) and several stays in Algeria I shared my experience with Mika. We had a lively discussion on the subject and in particularly on the difficulty to make a comprehensive research on this topic for Africa since the culture of the Maghreb countries, Soudan and other Islamic states are very different from the other States.

Kazuki, whom I met already a few years ago, is doing research on the rights of robots. We had a follow up discussion on the Civil Law Rules on Robotics as approved by the European Parliament.

During these discussions I could sense the difficulties they have in writing their doctoral thesis. It is true that it is not always obvious where to start and how to develop in a logic manner the results of their research. Therefore, using Kazuki’s research as an example I presented a methodology that I established over the years and which is based on the reasoning of Euclid’s elements of geometry.

A few days later we met again to visit some places of interest in Paris. Both Yorika and Mika were interested in bookshops specialized in legal issues but before going there we visited the Sorbonne where Professor Tatsuzawa (their mentor) studied for many years.

We thereafter went to the oldest legal bookshop in Paris: “Pedone”. It was founded in 1837, in 1887 it moved to its current location 13 rue Soufflot, Paris, in the middle of le Quartier Latin. It is without any doubt the best bookshop in Paris specialized in books on international legal issues. By entering the bookshop, I said that they were walking in the footsteps of Professor Tatsuzawa! Mika bought some books for her thesis.

 

Before entering the Sorbonne University Cujas building we went to see the Bibliothèque Sainte Geneviève.

We were a little tired and went to one of the terraces of the rue Soufflot for an espresso.

It was time to ask my, by now, classic question: “do you have a preferred mathematical formula or theory?”

Yorika and Mika would like to sleep on it before giving an answer as to Kazuki he said that he loves “amicable numbers”.

I was somewhat surprised by Kazuki’s choice, I only vaguely remembered amicable numbers. He told me that he has a very good friend who likes mathematics since high school. He taught him something about mathematics and it was him who for the first time mentioned the mystery of numbers. I thereafter discovered the sense of amicable number in a book written by Yoko Ogawa: 博士の愛した数式  (Hakase no aishita sūshiki – The Doctor’s beloved formula ). In English “The Housekeeper and the Professor”. This is an excellent book on the evolution through mathematics of a relation between a housekeeper, her son and a math professor. I often refer to this book in relation with Euler’s formula. If I remember correctly both professor Satoko Kawamura  and Kazuki recommended this book to me.

Instead of explaining what amicable numbers mean I prefer to quote from Ogawa’s book; a more poetic approach!

“I stopped washing and nodded, not wanting to interrupt the Professor’s first real attempt at conversation. 

“Your birthday is February twentieth. Two twenty. Can I show you something? This was a prize I won for my thesis on transcendent number theory when I was at college.” He  took off his wrist watch and held it up for me to sec. It was a stylish foreign brand, quite out of keeping with the Professor’s rumpled appearance. 

“It’s a wonderful prize,” I said. 

“But can you see the number engraved here?” The inscription on the back of the case read President’s Prize No. 284. ”Does that mean that it was the two hundred and eighty-fourth prize awarded?” 

I suppose so but the interesting part is the number 284 itself. 

“Take a break from the dishes for a moment and think about these two numbers: 220 and 284. Do they mean anything to you?” 

Pulling me by my apron strings, he sat me down at the table and produced a pencil stub from his pocket. On the back of an advertising insert, he wrote the two numbers, separated strangely on the card. 220                     284 

”Well, what do you make of them?” 

I wiped my hands on my apron, feeling awkward, as the Professor looked at me expectantly. I wanted to respond, but had no idea what sort of answer would please a mathematician. To me, they were just numbers. 

«Well . . . , “ I stammered” “I suppose you could say they’re both three-digit numbers. And that they’re fairly similar in size—for example, if I were in the meat section at the supermarket, there’d be very little difference between a package of sausage that weighed 220 grams and one that weighed 284 grams. They’re so close that I would just buy the one that was fresher. They seem pretty much the same—they’re both in the two hundreds, and they’re both even—“ 

“Good! « he almost shouted, shaking the leather strap of his watch. I didn’t know what to say. «It’s important to use your intuition. You swoop down on the numbers, like a kingfisher catching the glint of sunlight on the fish’s fin.” He pulled up a chair, as if wanting to be closer to the numbers. The musty paper smell from the study clung to the Professor.

“You know what a factor is, don’t you?” 

“I think so. I’m sure I learned about them at some point. . . ”For 220 is divisible by 1 and by 220 itself, with nothing leftover. So 1 and 220 are factors of 220. Natural numbers always have 1 and the number itself as factors. But what else can you divide it by?” 

“By 2, and 10. . .»Exactly! So let’s try writing out the factors of 220 and 284, excluding the numbers themselves. Like this.”

220 : 1 2 4 5 10 11 20 22 44 55 110

284 : 1 2 4 71 142 

The Professors figures, rounded and slanting slightly to one side, were surrounded by black smears where the pencil had smudged. 

“Did you figure out all the factors in your head?” I asked. 

“I don’t have to calculate them—they just come to me from the same kind of intuition you used. So then, let’s move on to the next step,” he said, adding symbols to the lists of factors.

220:1+ 2 + 4 + 5+10+11+20 + 22 + 44 + 55+11 0

284: 142 + 71 + 4 + 2 + 1 

“Add them up, “he said. “Take your time. There’s no hurry.” 

He handed me the pencil, and I did the calculation in the space that was left on the advertisement. His tone was kind and full of expectation, and it didn’t seem as though he were testing me. On the contrary, he made me feel as though I were on an important mission, that I was the only one who could lead us out of this puzzle and find the correct answer. 

I checked my calculations three times to be sure I hadn’t made a mistake. At some point, while we’d been talking, the sun had set and night was falling. From time to time I heard water dripping from the dishes I had left in the sink. The Professor stood close by, watching me. 

“There,” I said. “I’m done.” 

220 : 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

284 = 142 + 71 + 4 + 2+ 1 = 220 

“That’s right! The sum of the factors of 220 is 284, and the sum of the factors of 284 is 220. They’re called “amicable numbers”, and they’re extremely rare. Fermat and Descartes were only able to find one pair each. They’re linked to each other by some divine scheme, and how incredible that your birthday and this number on my watch should be just such a pair.” 

We sat staring at the advertisement for a long time. With my finger I traced the trail of numbers from the ones the Professor had written to the ones I’d added, and they all seemed to flow together, as if we’d been connecting up the constellations in the night sky.” (Emphasis added by me)

The next visit was Sorbonne Cujas building, we walked through the corridors where a lot of activities were going on, including the making of a movie. We entered a large classroom; some students were sitting there revising their lectures.   I invited Kazuki to go to the blackboard (in fact green board) and explain to us the theory of the amicable numbers. He first hesitated. “Is this allowed writing on the board” he asked me. I answered – “I do not know the only thing that can happen is that they kick us out!” Not really reassuring but he went and started his explanation.