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Ernst Sejersted Selmer

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Born
11 February 1920
Baerum, Oslo, Norway
Died
8 November 2006
Ski, Norway
Summary
Ernst Selmer was a Norwegian number theorist who also worked on mathematical cryptography. He is remembered for the Selmer group which played a part in the proof of Fermat's Last Theorem.
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Biography

Ernst Selmer was the son of Ernst Westerlund Selmer (1890-1971) and Ella Sejersted (1895-1968). Ernst Westerlund Selmer was born on 23 April 1890 in Funbo, Sweden but completed his secondary education at Kristiania Cathedral School. Kristiania is now Oslo, Norway. After graduating with a degree in German, he studied in Oxford, Montpellier, Leipzig and Marburg before being a research assistant in Hamburg from 1915 to 1917. He was a research fellow at the Royal Frederick University from 1917 to 1924. This University is now the University of Oslo. On 19 March 1919 he married Ella Sejersted, born 25 August 1895 in Oslo, Norway. Ernst and Ella Selmer had three children: Ernst Sejersted Selmer, born 20 February 1920, the subject of this biography; Nicolay Sejersted Selmer, born 6 April 1921; and Knut Sejersted Selmer, born 7 November 1924.

Nicolay Selmer, known as Nico, was an excellent mathematics student and won the Crown Prince Olav's Mathematics Prize for high school students. He began studying at the Norwegian Technical University in 1939 the year World War II began. He was involved in fighting in the spring of 1940 when Germany invaded Norway, then was in the resistance in Trondheim. In September 1941 he fled from Norway to Sweden, then continued to England and on to Canada where he began training as a fighter pilot. He died on 9 January 1943 when his plane crashed on a training flight.

Knut Selmer graduated with a law degree in 1949, was a deputy judge in Nord-Troms and Fredrikstad, 1949-1952, then, after serving as a research fellow at the University of Oslo from 1953 to 1959, was appointed as a professor of insurance law at the University of Oslo in 1959 where he remained until 1989. He was elected to the Norwegian Academy of Science and Letters from 1961 and was decorated as a Knight, First Class of the Royal Norwegian Order of St Olav in 1993. He died on 25 March 2009 in Oslo.

The Selmer family moved to several different houses in Oslo when Ernst, the subject of our biography, was young. They lived in the Fagerborg district until 1923, then, after a short stay in Asker, went to live in an area between Stabekk and Lysaker to the west of Oslo where there were detached houses with large gardens owned by professional people. Ernst attended Stabekk municipal secondary school which had been established in 1923. His younger brothers attended the same school. At this school Ernst had two favourite subjects, mathematics and chemistry. He was an editor of the school's mathematics magazine, Tall og tanker (Numbers and Thoughts), and in 1937 won the prize for the best mathematics student although in competition with pupils a year above him. In 1938 he won the Crown Prince Olav's Mathematics Prize for high school students.

At the secondary school he was also involved in sports, playing tennis, shot put, discus and javelin. He was good at the throwing events and in his final year at the school he was selected for Oslo's junior district javelin team. During these years at secondary school Selmer met and became close friends with the girl he would marry. The Selmer family had a country house in Hvitsten just south of Drøbak, where they spend most of their summers. Alf Faanes, the head of the bureau at the Ministry of Health, also had a country house in Hvitsten and Selmer became friendly with his daughter Signe Randi Johanne Schøning Faanes (1921-2011); she was known as Lillemor Faanes. They became engaged in 1939 and were married in 1945.

In 1938 Selmer entered the University of Oslo where he studied mainly mathematics but also mechanics, physics and chemistry. Life in Norway changed dramatically, however, when Germany invaded in the spring of 1940. By June 1940 Germany were in control of the country and a pro-German government ruled Norway. The rector of the University of Oslo was imprisoned and Adolf Hoel, a Norwegian Nazi Party member, was put in charge.

At the University of Oslo Selmer met Nils Berge Stordahl (1918-1984) and the two became friends. Stordahl became involved in Milorg, the main military resistance organisation in Norway which was set up in May 1941. One of the tasks that Milorg undertook was gathering intelligence and sending this to England. Selmer played a part in this highly dangerous work by encrypting messages which were to be sent to London, England. He worked as a student during the day, spending much time in the evenings doing the boring work of encrypting. The messages were then given to loyal railway workers on the Oslo to Stockholm trains and the messages were then forwarded from Stockholm to Norwegians in exile in London.

This encrypting work was time consuming but Selmer found time to undertake number theory research and publish several papers. He had read the book Introduction to the theory of numbers (1938) by G H Hardy and E M Wright and began studying the open questions posed in it. He was particularly interested in making hand calculations to investigate the open problems, publishing his results in both Norwegian and in German. For example in 1942 he published: Trigonometrisk løsning av en tredjegradsligning med tre reelle røtter Ⓣ (1939); En enkel summasjonsmetode i primtallsteorien, og dens anvendelse pa "Bruns sum" Ⓣ (1942); Om sannsynligheten for at et tall er et primtall Ⓣ (1942); Eine neue hypothetische Formel für die Anzahl der Goldbachschen Spaltungen einer geraden Zahl, und eine numerische Kontrolle Ⓣ (1942); Tafel der Zwillingsprimzahlen bis 200.000 Ⓣ (1942); and Die Goldbachschen Zwillingsdarstellungen der durch 6 teilbaren Zahlen 196.302 - 196.596 Ⓣ (1942). The last two of these are joint with Gunnar Nesheim. The twin primes paper tabulates those values of n≤200000n ≤ 200000 for which both 6n+16n+1 and 6n−16n-1 are primes. The numbers of prime pairs with n≤100000n ≤ 100000 is 1224 that there are 936 with 100000≤ n ≤200000100000 ≤ n ≤ 200000. The table is calculated with hand calculations. In 1943 he published: To rekker for summen √x=Σp≤xln⁡p√x = \Sigma _{p} ≤x \ln p Ⓣ (1943); and Eine numerische Untersuchung über die Darstellung der natürlichen Zahlen als Summe einer Primzahl und einer Quadratzahl Ⓣ (1943).

In the autumn of 1943 Selmer heard that the German Gestapo were planning to raid the University of Oslo. He certainly knew that with his resistance activities, his life would be in extreme danger if he were captured in such a raid. He left the university and was not there when the Gestapo raid took place on 30 November 1943. The Gestapo arrested male staff and students and most were sent to concentration camps; the University was closed. Selmer aimed to escape to Sweden and, having orienteering experience, made most of the journey on foot. It was a long and very strenuous journey much of it through dense forests but, on 11 February 1944, he arrived at the refugee reception centre at Kjesäter Manor in Vingåker, Södermanland. After 3-4 days of medical examinations and interrogations he was provided with new clothes, a passport and some money. His first thought was to continue his study of mathematics at the University of at Uppsala but, given his experience with cryptography, he made contact with his friend Stordahl, who had escaped to Sweden before Selmer, and he was sent to Luntmakargatan in Stockholm to take an advanced encryption course. While there he studied the Hagelin cipher machine, designed by Swedish cryptographer Boris Hagelin.

After his studies in Stockholm, Selmer was sent to London, England, in the summer of 1944 where he was given responsibility for the technical maintenance of all the Hagelin machines in use in the Norwegian cipher council. He arrived in London in June 1944, the month the Wehrmacht began launching the V-1 flying bombs against London. In an interview in March 2021 Selmer recalled the horror (see, for example, [5]):-
I arrived in London just at the same time as the flying bombs, so it was quite dramatic. This was the V-1's and they had just started falling all over London when I arrived. ... When we heard one of these bombs stop its engine above us, we just had to duck because of the splinters and bits and pieces. I was never very close, but once I was at least close enough that in the boot of the car we found an aluminium splinter from a flying bomb. That's how close we were.
The Norwegian cipher team collaborated closely with the British. Selmer was responsible for both technical maintenance of the Hagelin C-38s, the rotor-based encryption machines, and for the encoding and decoding of messages. Selmer was able to address weaknesses in the encryption and make the messages more secure. Although the Germans were able crack these codes during the war, it took them many hours to decode each message by which time the knowledge gained had little consequence.

When the German occupation of Norway ended in May 1945 [10]:-
I was sent to Tromsø, in Northern Norway as a cipher officer. But there was no need for cipher, so I spent a fabulous summer under the midnight sun. I managed to get hold of some of the famous German Enigma cipher machines, and took them with me to the military headquarters in Oslo.
Selmer then returned to his studies at the University of Oslo, which had reopened, where he majored in mathematics and was awarded the degree of Candidate of Mathematics and the Natural Sciences in 1945 with a remarkable grade point average. Back in Oslo, he married Lillemor Faanes in 1945. Helge Tverberg, in his memorial speech for Selmer, said [17]:-
He was also able to marry Lillemor Faanes who had been waiting for him at home. She was his great support throughout his life and his great efforts in many fields would probably not have been possible without her.
Ernst and Lillemor Selmer had a daughter, Johanne-Sophie Selmer (born 1955). She was educated at Bergen Cathedral School, then studied at the University of Bergen and in the Department of Marine Microbiology at Gothenburg University. She was awarded her doctorate in 1988 from Gothenburg University for her thesis Ammonium regeneration in the marine environment. She became a microbiologist in the Department of Biology, Faculty of Social and Life Sciences, Karlstad University.

After graduating, with no immediate prospect of a university position, he became a mathematics teacher at Kristelig Gymnasium, a private Christian preparatory school located in Oslo. He did not enjoy the experience, becoming quite disappointed by the academic level of some of the high school students. In 1946 Ernst Selmer was appointed as a lecturer in mathematics at Oslo University. In the same year Nils Stordahl became head of the Cipher Department of the Norwegian Armed Forces Security Service and immediately hired Selmer as a consultant for the Cipher Department [10]:-
My first big task, and my most fantastic cryptological experience, was to establish a (hopefully) safe communication system for the Norwegian equivalent of MI5 and Scotland Yard Special Branch. We based it on the German Siemens teleprinter, with an additional unit for encryption/decryption.
Hans Morten Synstnes writes [19]:-
From 1946 he took the lead in an academic effort to protect secrecy and national sovereignty. Like other external paid consultants, he shared the Cipher Department's view that scientific progress was essential to national security.
Of course, at the time his work for the Security Service was secret. No one knew he had this second job so he was not allowed to declare his income from that work for tax purposes. (One never gets away without paying tax, of course, and it was deducted by the Security Service before he received his remuneration.)

Haanaes describes Selmer as a lecturer in [5]. Since:-
... the University of Oslo was closed for two years because of the war, Selmer has plenty to do. He writes exercise books on a continuous basis and lectures on basic mathematical topics for students who are going to become social economists. Many of them are only a few years younger than the lecturer.

Svein Nordbotten had Selmer as a lecturer in mathematics in 1948. He describes Selmer as scholarly, clear and quite entertaining, and as a man who likes to show off a bit. Selmer has a little circus trick that he occasionally performed after lectures, where he scanned the numbers on the board with a quick hand movement and gave the students their sum. It always works, and many are impressed by the lecturer's abilities as a calculating machine. Later in life, Selmer reveals to Nordbotten that his lectures were so systematically conducted that he had calculated the answer in advance.
Once back in Norway after war, Selmer resumed publishing papers. Now, however, he did not write papers in German but wrote mostly in English with some in Norwegian. For example, in 1946 he published four papers: An approximate formula for τ(z)\tau(z); Some rapidly converging series for the elliptic p-function of Weierstrass; A simple trisection formula for the elliptic p-function of Weierstrass in the equianharmonic case; and Noen tilnaermelsesverdier for π Ⓣ.

In 1949, Selmer received a scholarship and spent the spring semester in England at the University of Cambridge. There he worked with Louis Mordell and also became friendly with Ian Cassels who was completing the work for his Ph.D. thesis advised by Mordell. It was Cassels who named the Selmer group in 1962 when working on rational points on cubic curves, writing in his paper Arithmetic on curves of genus 1. III. The Tate-Šafarevič and Selmer groups:-
We shall call it a Selmer group because Selmer initiated the present work.
The Selmer group has played a large role in Andrew Wiles' proof of Fermat's Last Theorem.

Atle Selberg, a Norwegian mathematician known for his work in analytic number theory, became a permanent member of the Institute for Advanced Study (IAS) at Princeton in 1949 and in 1950 was awarded a Fields Medal. The minutes of the meeting of the School of Mathematics of the IAS of 26 September 1950 contain the following (see, for example, [5]):-
Professor Selberg raised the question of Dr Ernst Selmer, a Norwegian mathematician, who is interested in the computer project and would like to spend six months here as a member. Such an application would be welcomed when and if it is submitted.
In fact although Selmer had submitted considerably more papers than would be required for a Ph.D., in fact he had not, at that point, submitted a thesis. Only mathematicians with a doctorate would be welcomed at the IAS but whether Selberg's reference to "Dr Selmer" is a deliberate mistake or a genuine error is of little consequence. Selmer was awarded a Rockefeller Foundation Fellowship for 1951-52 as a visiting researcher, first at the IAS, then at Berkeley University.

Ernst and Lillemor Selmer sailed on the MS Oslofjord from Oslo to New York arriving on 16 January 1951. Reaching Princeton on Saturday afternoon, the first person he saw he believed was the janitor and asked him if he would show him round the IAS. The "janitor" was happy to oblige; only after his tour did Selmer realise he had been shown round by Albert Einstein [5]:-
Selmer likes to talk about this experience when he returns to Norway. The fact that Lillemor, a few months into his stay, is a hair's breadth away from running over Einstein during a practice drive on the Institute's grounds also gradually becomes a good story.
At the IAS a computer was being constructed for John von Neumann. It was being built under von Neumann's direction and construction had begun in 1946. Selmer is recorded as being a member of the School of Mathematics of the IAS from January to June 1951 and as a visitor to the Electronic Computer Project from January to December of 1951. He worked with the Computer Project team and the IAS computer was in limited operation by the summer of 1951. It became fully operational by the summer of 1952. Selmer is also recorded as being a member of the School of Mathematics of the IAS from January to June 1952. He was not at Princeton for the whole of these eighteen months since in the middle of 1951 he travelled to Berkeley University where the California Digital Computer (CALDIC) was being constructed by Paul Morton and D H Lehmer. Selmer and his wife had saved up enough money to buy a car and drove to California in their 1940 Packard 160 Convertible. Work on the CALDIC computer was just beginning and Selmer designed the add-subtract control for the computer.

The Consolidated Electrodynamics Corporation (CEC) was about to begin work on the design of the Datatron computer in late 1951 and were looking for experts to assist in the design. They approached von Neumann asking for him to recommend possible consultants and he suggested Selmer. He was employed as a consultant by CEC management in late 1951 and during the following six months designed most of the logic for the Datatron [10]:-
I undertook the job, which was finished after my return to Norway. I actually did the design down to every single tube (no transistors existed then). The company wanted a completely decimal machine, with a magnetic drum as its main memory. After a while, the company got economic problems and was swallowed by Burroughs, who entered the computer race with "my" machine Burroughs 205. In the late 1950s, this was the most serious competitor to the famous IBM 650. My machine was perhaps the only, and certainly the last, larger computer where the complete logical design was a "one man job".
In a letter dated 27 September 1951, von Neumann confirms that Selmer is to be employed on the computer project at Princeton with a monthly salary of $450. Early in 1952 the Selmers drove back to Princeton via Florida and the Everglades, reaching Princeton on 22 February 1952. Selmer worked on computer projects but also continued his research on Diophantine equations for his Ph.D. thesis. On 13 June 1952 the Selmers sailed from New York on their way back to Oslo. This did not mean that Selmer stopped contributing to the development of American computers [5]:-
Construction drawings for Datatron, formulas, tables, problem descriptions, proposed solutions, questions, counter-questions and the occasional answer with two lines underneath are shuttled between Oslo and Pasadena. In some places one can read between the lines that Selmer's ability to take unwavering positions slightly on the edge of his actual field of expertise irritates the Americans a little. Since the exchange of letters continues, Selmer's contributions must nevertheless be seen as very useful.
Selmer had completed his Ph.D. thesis The Diophantine Equation ax3+by3+cz3=0ax^{3} + by^{3} + cz^{3} = 0 while in Princeton and submitted it for publication in Acta Mathematica. It was published in 1951, see [9]. Selmer was awarded his Ph.D. by the University of Oslo in 1952. In his thesis he studied ax3+by3+cz3=0ax^{3} + by^{3} + cz^{3} = 0 where a,b,ca, b, c are integers and he sought solutions (x,y,z)(x, y, z) in integers [17]:-
He solved many equations and showed that many others had no solutions. The case a=3,b=4,c=5a = 3, b = 4, c = 5 is particularly interesting. The corresponding equation has no solution despite passing certain tests. This shows that the so-called Hasse-Minkowski principle, which applies to quadratic equations, does not apply to cubic equations. The example will forever link Selmer's name to the triple 3, 4, 5, just as Pythagoras's is linked to the same triple.
Selmer continued to publish, a 1954 paper continuing the work of his Ph.D. theses. In this paper he solves certain special cases left unsolved in [9] by using the electronic computer at the Institute for Advanced Study at Princeton. Ian Niven writes in a review [18]:-
The author continues his studies of the equations (1) x3+y3=Az3x^{3} + y^{3} = Az^{3} and (2) ax3+by3+cz3=0ax^{3} + by^{3} + cz^{3} = 0. The tables of basic solutions of equation (2) are completed for all positive cube-free abc ≤ 500, with extensions beyond 500 in some cases. Part of this was done by work with an electronic computer. These tables, together with cases proved unsolvable by the method of J W S Cassels enable the writer to complete the solutions of (1) for all positive cube-free A≤500A ≤ 500.
In 1957 Selmer left Oslo to become professor of mathematics at the University of Bergen. It was a position he would continue to hold until he retired in 1990. He gave his inaugural lecture in Bergen in 1958 and used the occasion to stress what electronic computers could add to mathematics. He played a large role in the development of mathematics teaching at the University of Bergen and was the main architect for the new curriculum for the Faculty of Mathematics and Natural Sciences introduced in 1959. He was elected vice-dean of the Faculty in 1960 and served in this role until 1966 when he became dean of the Faculty, serving until 1968.

He took a break during his period as vice-dean, spending the academic year 1964-65 in England at the University of Cambridge. There he worked on linear recursion and periodic sequences, publishing the paper Binaere registre og lineaer rekursjon in 1968. The paper contains the following summary:-
It is shown how binary feedback shift registers lead to binary linear homogenous recurrence relations. The structure of the solution space for such a relation is treated in detail by Zierler's method. The more difficult distribution problems are only mentioned briefly.
This work in a sense bridged the gap between the number theory research he had undertaken and his work with the Cipher Department on coding.

Selmer was elected to the Norwegian Academy of Sciences and Letters in 1961. He was a member of the Council for Electronic Data Processing from its establishment in 1961 to 1973. He was made a Knight of the Order of St Olav, 1st class, in 1983.

Among his many activities, he was the editorial secretary of the journal Nordisk Matematisk Tidskrift. He took over this position from the University of Oslo mathematician Ingebrigt Johansson (1904-1987) in 1954 and continued in this role until 1978. This was a job that involved a very considerable amount of work as is evident by the fact that, when he retired from the role, a group of eight mathematicians in Kristiansand replaced him.

For more details of Selmer's contributions after he was appointed to the University of Bergen, see our translation of his Memorial by Helge Tverberg at THIS LINK.

Selmer retired in 1990 and went with his wife to live in Ski, Norway. He continued to have good health until the autumn of 2004 when he suffered a stroke. He never fully recovered and died in November 2006. He was buried in Hobøl Cemetery, Bjerve, Hobøl municipality, Østfold county. The picture of his grave on the website "Find a grave" gives his date of birth as 11 February 1920 although confusingly the site itself gives 20 February 1920. Some other sites also give 20 February but Selmer himself states his birthday in on 11 February. His wife Lillemor died in Ski on 26 January 2011.
Other Mathematicians born in Norway
A Poster of Ernst Selmer

References (show)

  1. Bak ryggen på sjefene knekket professoren koder for Forsvaret, Bergens Tidende (11 January 2021)
    https://www.bt.no/nyheter/lokalt/i/pAdn2E/bak-ryggen-paa-sjefene-knekket-professoren-koder-for-forsvaret
  2. Ernst Sejersted Selmer, Mathematics Genealogy Project (2025).
    https://www.genealogy.math.ndsu.nodak.edu/id.php?id=319726
  3. Ernst (Lill-Ernst) Sejersted Selmer, ancestry.com (2025).
  4. Ernst S Selmer, Institute for Advanced Study (2025).
    https://www.ias.edu/scholars/ernst-s-selmer
  5. Ø R Haanaes, Professor i hemmelig tjeneste. Matematikeren, datapioneren og kryptologen Ernst Sejersted Selmer (Bergen, 2020).
  6. Ø R Haanaes, 100 år siden personnummerets far ble født, forskning.no (11 February 2020).
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  8. T J Sawyer, Tom's Datatron 205, tjsawyer.com (2025).
    https://tjsawyer.com/B205Home.htm
  9. E S Selmer, The diophantine equation ax3+by3+cz3=0ax^3 + by^3 + cz^3 = 0, Acta Mathematica 85 (1951), 203-362.
  10. E S Selmer, From the Memoirs of a Norwegian Cryptologist, in T Helleseth (ed.), EUROCRYPT 1993, Lecture Notes in Computer Science 765 (Springer, Berlin, Heidelberg, 1993), 142-150.
  11. E S Selmer, Personnummerering i Norge. Litt anvendt tallteori og psykologi, Nordisk Matematisk Tidsskrift 12 (1964), 36-44.
  12. E S Selmer, Registration Numbers in Norway: Some Applied Number Theory and Psychology, Royal Statistical Society. Journal. Series A: General 130 (2) (1967), 225-231.
  13. K Stordahl, Datapioneren Ernst Selmer gjorde Norge til en stormakt innen kryptering. Har fått et norsk forskningsenter oppkalt etter seg, DIGI (5 May 2019)
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  14. K Stordahl, "Frederic" var Vest-Europas kraftigste datamaskin, og den var norskeid, DIGI (17 April 2019)
  15. H K Strand, Arven etter Selmer, Khrono (23 January 2021).
    https://www.khrono.no/arven-etter-selmer/541869
  16. H Twerberg, Nekrolog Ernst Sejersted Selmer, Aftenposten (1 December 2006).
    http://selmer-norway.no/showmedia.php?mediaID=902
  17. H Twerberg, Minnetale over professor Ernst Sejersted Selmer, Norwegian Academy of Science and Letters (7 June 2007).
    https://depotbiblioteket.no/cgi-bin/m2?tnr=1557697
  18. I Niven, Review: The diophantine equation ax3+by3+cz3=0ax^3 + by^3 + cz^3 = 0. Completion of the tables, Mathematical Reviews MR0067131 (16,674e).
  19. H M Synstnes, Den innerste sirkel. Den militære sikkerhetstjenesten 1945-2002 (Dreyers forlag, 2016), 406.

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Written by J J O'Connor and E F Robertson
Last Update December 2025