# Jan Popken

### Quick Info

Hijkersmilde, near Smilde, Netherlands

Amsterdam, Netherlands

**Jan Popken**was a Dutch mathematician specialising in number theory.

### Biography

**Jan Popken**'s parents were Jan Popken (1868-1954), son of Pieter Popken and Annis Boerhof, and Jantje Hofman (1873-1946), daughter of Jan Hofman and Janna Haveman. Jan, the subject of this biography, had three siblings Annis Popken, Willem Popken and Pieter Popken. We have given his birthplace as Hijkersmilde near Smilde which was the situation when he was born but we note that Hijkersmilde is now a district of Smilde. The family were well off with Jan Popken Sr. having a thriving farm. He was also a threshing entrepreneur and, at the height of his business, had two threshing machines at his disposal and a hundred workers.

After attending secondary school, Popken was able to study mathematics and physics at the University of Groningen since his parents were well off and able to support their son financially through university. Popken was taught by Johan Antony Barrau (1873-1953) who had been advised by Diederik Korteweg for his doctorate. Barrau had taught at the University of Delft before being appointed professor of synthetic, analytical and descriptive differential geometry at Groningen in 1913. Another of Popken's teachers, who arrived in Groningen in 1923, was the number theorist Johannes Gaultherus van der Corput (1890-1975). Popken graduated with a candidate's degree (the candidate's degree in the Netherlands is a first degree of the standard of a B.Sc.) on 15 February 1927. Popken continued to study at Groningen, undertaking research, and obtained a second degree cum laude on 2 July 1930. He had already some papers published in both French and German, namely

*Sur la nature arithmétique du nombre e*Ⓣ (1928) and

*Zur Transzendenz von e*Ⓣ (1929).

Let us try to explain the problem on which Popken was undertaking research. A number $k$ is

*algebraic*if, for some polynomial $f(x)$ with integer coefficients, not all zero, $f(k) = 0$. If $k$ is not algebraic then it is called

*transcendental*so, in that case $f(k) ≠ 0$ for every such polynomial $f$. Liouville showed the existence of transcendental numbers in 1844 and constructed such numbers. The first number, not specially constructed to have this property, which was shown to be transcendental was $e$ by Hermite in 1873. Now define the

*height*$H$ of a polynomial to be the sum of the absolute values of its coefficients. For a given $n$ and $H$ there are only finitely many polynomials $f$ with degree ≤ $n$ and height ≤ $H$. If $k$ is transcendental, then for a given $n$ and $H$ there must exist $C$ such that $|f(k)| > C$ for every $f$ of degree ≤ $n$ and height ≤ $H. C(n, H)$ is called the

*transcendental measure*of $k$. The problem of finding the transcendental measure of e was posed by Émile Borel in 1899 and, in 1928, soon after he had completed his first degree, Popken read Borel's paper and decided to try to improve the bound for the transcendental measure of e. That is precisely what he did in the 1929 paper we mentioned above.

After graduating in July 1930, Popken spent two semesters at the University of Göttingen with the number theorist Edmund Landau. The result of the expertise that Popken gained from Edmund Landau was seen in his joint paper with Jurjen Koksma,

*Zur Transzendenz von eπ*Ⓣ which was published by Crelle's Journal in 1932. In this paper the authors gave a transcendental measure for $e^{\pi}$. After the study trip to Göttingen, Popken returned, for financial reasons back to his birthplace Smilde. These were difficult times, even for such a gifted young mathematician, and jobs were scarce. Withdrawn to his parents' home, he wrote his thesis

*Über arithmetische Eigenschaften analytischer Funktionen*Ⓣ which earned him a doctorate from the University of Groningen on 12 July 1935. His thesis advisor was Johannes van der Corput who wrote the following about Popken's thesis in [4] (we give a full quote):-

In my opinion, this dissertation is one of the best in mathematics to have appeared in the Netherlands in this century. If, after exactly 36 years, I read it again, and delve into the subject again, I am struck not only by the results achieved and by the deep reasoning used, reasoning different from that usually used in the rest of mathematics, but also by the graceful form in which he has cast his work. Although many eminent mathematicians had devoted their energies to this difficult area, only a few general propositions were known, while it was the intention of the author to derive a number of comprehensive propositions. To this end, he divides his thesis consisting of 121 pages into six parts. In the introduction, he discusses the old results obtained by others and the new ones from the author himself. Five parts follow. Parts I and II are devoted to the solutions of Hurwitz differential equations; part III deals with algebraic functions, part IV with functions that satisfy algebraic equations and finally part V with certain functions that he calls elementary.Van der Corput felt that Popken never received the credit that he deserved for this remarkable piece of work [4]:-

The dissertation is written in the well-known Edmund Landau style, nowhere a word too much, but nowhere a word too little. With regard to the propositions, people sometimes wonder what the point of a particular proposition is, but nowhere is there given any information on this, because that connection appears later in the proof of a major proposition where the earlier proposition is applied. In an oral presentation, when dealing with such an assertion, it will as a rule be indicated with a few vague words how it fits into the direction of the requested theorem, but that indication does not fit into the aforementioned style, where no vagueness is allowed.

In Popken's dissertation there are fifteen major propositions with almost twice as many, most elaborate, minor propositions, which, all artfully and ingeniously combined, yield exactly the intended results. Many mathematics practitioners can read the dissertation without difficulty, because the entire argument is divided into so many small steps that each result follows from a single carefully worded statement. But a reading, step by step, takes weeks and even though the reader can then guarantee the correctness of the results achieved, he does not yet oversee the series of thoughts that led the author when writing the work.

The period after his dissertation until the end of World War II was bleak for this scholar. His dissertation, however important, was actually read, understood and appreciated by only a few in the Netherlands. After all, mathematicians do not easily have the opportunity to take a period of one month away from the time available to them, solely for the purpose of studying a piece of writing that is outside of their interest. I have the impression that he has received a shock during that period, of which he never fully recovered, but he never spoke about it.From 1937 to 1945 he worked as a teacher in The Hague, Veendam, and in Ter Apel. In 1937 Popken habilitated at the University of Groningen with the thesis

*Over het rekenkundig karakter van getalen*Ⓣ. On 3 May 1937 he gave his public inaugural lecture entitled

*Over het rekenkundig karakter van getallen*Ⓣ. He remained a docent at the University of Groningen until 1940. The situation in the Netherlands changed dramatically in May 1940. From the start of World War II in September 1939 the Netherlands had declared itself neutral but on 10 May 1940 German troops invaded the country. Popken left Groningen and moved to the University of Leiden where he was appointed as a docent in 1940. The students, however, called for a strike after the Dean of the Faculty of Law made a protest speech on 29 November 1940. The German occupiers closed the university but students were allowed to remain until they had taken their examinations in November 1941. At this stage the Germans attempted to Nazify the university but most of the professors resigned in protest. Popken was still able to work as a school teacher but he had no university position until 1945 when the war ended.

After the war Popken was appointed as a scientific assistant at the University of Groningen, a position he held from 1945 to 1947. The war had caused a gap in Popken's publication record which had been an average of around two papers a year up to 1940. The war, however, forced a gap in the record with no papers appearing between 1941 and 1944. He was appointed to the Mathematical Centre in Amsterdam in 1947 but later the same year he was appointed as a professor at the University of Utrecht. He delivered his inaugural lecture

*De jeugdperikelen van het getal*Ⓣ on 20 October 1947.

We give an English version of Popken's lecture on the primitive peoples' difficulties with numbers at THIS LINK.

Popken had married in Groningen on 9 August 1947 to Catharina Cornelia Johanna ten Cate, the stepdaughter of van der Corput. Three children were born from that marriage, Jeannette Popken, born in Utrecht on 19 May 1948, Peter Popken, born in Utrecht on 30 November 1949, and Marion Popken, born in Utrecht on 15 December 1952.

On 19 August 1950 Popken arrived in New York having sailed on the

*Volendam*from Rotterdam, Netherlands. He was on his way to the International Congress of Mathematicians held at Harvard University from 30 August 30 to 6 September 1950. He sailed back from New York to Rotterdam on the same ship, the

*Volendam*, leaving on 18 September. In October 1950 the Board of the Koninklijke Wiskundig Genootschap (Dutch Mathematical Society) appointed a small committee with the task of drawing up a report on the structure and regulations for the International Congress of Mathematicians to be held in Amsterdam in 1954. H D Kloosterman was chair of this committee with Haantjes as its secretary and Popken as one of the three other members.

Popken held the position as professor at the University of Utrecht for eight years before moving to Amsterdam when he was appointed as Professor of analysis, algebra and number theory at the university there on 19 January 1955. He gave his inaugural lecture,

*Mathesis en maatschappij*Ⓣ, on 11 March 1957. J H Wansink writes [5]:-

The danger is not inconceivable that mathematical lectures are difficult to digest for some listeners due to their technical nature and are therefore only slightly appreciated by these listeners. The Amsterdam inaugural lecture of Prof dr Popken, also addresses the non-mathematicians in his audience in a richly documented argument. The work of the mathematician is to discover new paths in mathematics, paths that are sometimes decisive for the course of humanity, even though the mathematician may have been far ahead of his time. This statement is illustrated by compelling examples. The inaugural lecture also sketches how the tragic isolation and unpopularity of mathematics grew especially in the 19th century, at a time when many influential outsiders often identified mathematics with totally petrified school mathematics. A turn comes in the twentieth century; an understanding arises for the meaning of mathematics in philosophy. In the second industrial revolution we now live in, enthusiastic and successful searches are being made for new ways to which mathematics can be applied. Nobody left this lecture uninformed.Popken spent the summer of 1959 at the University of California in Berkeley. While at Berkeley, from 18 June to 20 July 1959, Popken delivered the course 'Elementary mathematics for advanced students'. This course studied selected topics in elementary mathematics, with particular emphasis on their historical development. It was designed to be particularly useful for high school teachers of mathematics. He also gave the course 'Algebraic and transcendental numbers'. It covered irrational numbers, algebraic numbers, approximation of algebraic numbers by rationals, Liouville numbers, the theorem of Thue-Siegel-Roth, transcendence of e and π, the Lindemann theorem, and some recent results in the theory. Note that Roth in the Thue-Siegel-Roth theorem is Klaus Roth.

On 31 August 1962 Popken, his wife Catharina, and his three children Jeannette, Peter and Marion flew to New York. They gave their permanent address as Boslaan 2, Amstelveen, Holland. We note that Amstelveen is about 8 km to the south of Amsterdam. Popken was on his way the spend time at the University of California in Berkeley and gave their address in the United States as 1597 Hawthorne Terrace, Berkeley. While in the United States, Popken participated in the 1962 conference 'Studies in mathematical analysis and related topics' held at Stanford University. He delivered the paper

*On multiplicative arithmetic functions*which was published in the proceedings of the conference.

On 12 April 1965 Popken changed from the position of Professor of Analysis, algebra and number theory to being Professor of Pure Mathematics, still at the University of Amsterdam.

In the later stages of his career, Popken became increasingly interested in the history of mathematics. He continued, however, to publish papers on a variety of different topics. For example, after being appointed to the University of Amsterdam he published papers such as:

*Some theorems concerning transcendental numbers*(1956);

*Arithmetical properties of the Taylor coefficients of algebraic functions*(1959);

*Algebraic dependence of arithmetic functions*(1962);

*On the so-called von Neumann numbers*(1962);

*Irrational power series*(1963);

*Note on a generalization of problem of Hilbert*(1965);

*Algebraic independence of certain zeta functions*(1966);

*A measure for the differential-transcendence of the zeta-function of Riemann*(1969); and

*A contribution to the Thue-Siegel-Roth problem*(1970).

This publication record hides the fact that in the final years of his life he was suffering a tiring and painful illness. Despite the extremely difficult conditions, he continued to work up to his death at 64 years of age.

Among the honours he received was election to the Royal Netherlands Academy of Arts and Sciences in 1954.

Finally, we note that at least seven of his Ph.D. students (some while he served at Utrecht and some at Amsterdam) became university lecturers in mathematics and all these seven have in turn supervised Ph.D. students of their own. The most famous of these seven is, perhaps, is Abraham van der Sluis (1928-2004) who graduated from the University of Amsterdam in 1956 with his thesis

*General Orthogonal Polynomals*and went on to set up the computer centre at the University of Utrecht where he was director until 1970.

### References (show)

- J Popken, Over het getal π,
*Euclides***18**1(2) (1941), 7-14. - Jan Popken 1905-1970,
*University of Groningen*(November 2018). http://www.math.rug.nl/bernoulli/Geschiedenis/Popken - J Popken, De jeugdperikelen van het getal,
*Euclides***23**(2) (1947-48), 80-97. - J G van der Corput, Levensbericht J Popken, in
*Royal Netherlands Academy of Arts and Sciences Jaarboek, 1970*(Amsterdam, 1970), 240-245. - J H Wansink, Mathesis en Maatschappij door Prof Dr J Popken,
*Euclides***32**(1956-57), 301.

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Written by J J O'Connor and E F Robertson

Last Update November 2019

Last Update November 2019