# Roger Apéry

### Quick Info

Rouen, France

Caen, France

**Roger Apéry**was a French mathematician best known for proving that ζ(3) is an irrational number.

### Biography

**Roger Apéry**'s father, Georges Apéry (1887-1978), was born in Constantinople in 1887 but he was of Greek origin. He was a student at the École Nationale Supérieure d'Ingénieurs at Grenoble having arrived in France a couple of years earlier, in 1903, to prepare for his studies. He was still in France when World War I broke out in 1914 and, as a way of gaining French citizenship, he enlisted to fight for France. For his service he was awarded the Croix de Combattant Volontaire. While fighting in the Dardanelles, he contracted typhoid and was returned to France to recuperate. He moved to Rouen where he married Justine Vander Cruyssen (1992-1965). She was a piano teacher and, disliking her Flemish name, she was known by the French equivalent of Vander Cruyssen, namely Delacroix. She also disliked Justine and was known as Louise. Roger, the subject of this biography, was their only child.

The Apéry family spent the first four years of Roger's life in Rouen then, in 1920, they moved to Lille where Georges worked as an engineer. Roger attended the lycée Faidherbe in Lille and he was such a bright child that by 1926 he was two grades ahead of children of his age. It was in 1926 that the family moved again, this time going to Paris where Roger continued his education at the lycée Ledru-Rollin. Again he showed his brilliance, being particularly strong in history and mathematics. It was at this school that, in 1928, he first met Euclid's axioms and became fascinated by the parallel postulate. Perhaps even at this stage the passion that he had for geometry throughout his life became evident. The Great Depression, which began in the United States in the summer of 1929, soon began to impact other countries and by early 1930 France was in recession. The Apéry family had moved into very basic accommodation in Paris with the intention of improving their position as soon as their finances allowed them to better themselves. It was not to be, however, for the economic downturn in France led to Georges Apéry loosing his position as an engineer. Although he was only 43 years old, he never managed to get a job as an engineer again being judged too old by the time the French economy improved. He was employed as a watchman at the Ministère des Anciens Combattants and his wife helped out with the family finances by giving piano lessons. However, they never managed to afford better accommodation.

Roger, however, continued to show his brilliance, moving to the famous lycée Louis-le-Grand in Paris. Because their home was not suitable for anyone to study in, Roger boarded with the monks of the École Bossuet (named after Jacques-Bénigne Bossuet, a Roman Catholic bishop known for his literary works). One day when Apéry was not in his room, the abbot searched among his papers and found one which contained the words, "Molière, who was right not to like priests ...". Apéry was given a punishment detention, and from this time on he developed a dislike of priests and the Church. He continued to be fascinated by mathematics, however, and in 1932 he met the cross-ratio as a projective invariant; his love of geometry became more specific, turning into a love of algebraic geometry. Apéry was a Concours Général laureate in 1932, coming third in mathematics in the whole of France in the junior competition. In the following year he went one better, coming second in mathematics in the senior Concours Général. He also received an honourable mention in physics. He was awarded his baccalaureate in mathematics and philosophy in 1933 and prepared for university entry.

A financial scandal came to light in December 1933 when investments in Alexandre Stavisky's credit organisation became worthless. Stavisky was found dead in January 1934 and the official story that he had committed suicide was not believed by right-wing supporters who believed in a government conspiracy. Riots followed outside the Chamber of Deputies on 6 February and Apéry, who was already deeply politically aware, took a strong stand for radical socialism opposed to the right-wing demonstrators. In April 1934, he joined Camille Pelletan's Radical Party. This was to have consequences for his academic career for, despite a brilliant performance in mathematics, Apéry was not ranked sufficiently highly to gain entry to the École Normale Supérieure in 1935. Having worked hard to improve his performance in chemistry, he was ranked second in the admission examination for the École Normale Supérieure in the following year and began his university career.

The Munich agreement of September 1938, allowing Germany to annex the Sudetenland, was signed by the French premier Édouard Daladier, a member of the Radical Party. This prompted Apéry, who was still an undergraduate, to resign from the Radical Party in protest. Apéry would continue as a staunch socialist but outside the mainstream parties. At the École Normale Supérieure he was coached by Raymond Marrot and came first equal in the Agrégation de Mathematiques in 1939. Marrot was a Communist and the two men became friends sharing a passion for left-wing politics. The other candidate who came first equal with Apéry was Jacqueline Ferrand, better known by the name she adopted after her marriage, Jacqueline Lelong-Ferrand. Jean Dieudonné was an examiner for analysis and he wrote:-

I was a member of the agrégation jury, for the only time in my life, by the way, and I gave a rather unusual analysis problem. Only two of the papers impressed me with their sense of analysis and precocious maturity very rare among candidates for the agrégation. Those two were by Roger Apéry and Jacqueline Ferrand.Apéry was already undertaking research and published

*Sur les sextiques à 8 rebroussements*Ⓣ (1939).

In August 1939, Russia and Germany signed a secret pact, the so-called Ribbentrop-Molotov pact, to divide Poland between them. The two-pronged attack on Poland began on 1 September 1939 and, on the following day, Britain, France and several other countries, declared war on Germany. The Germans attacked from the west and the Russians from the east, quickly defeating the Polish army. Apéry was drafted into the French army on 16 September 1939. Over the following months France was not involved in any fighting, but spent time strengthening the Maginot line, designed to protect the country from an invasion by Germany. In February 1940 Apéry was promoted to sublieutenant in the 145th Artillery and sent to Nancy. The war changed dramatically for France on 10 May 1940 when the German army crossed the Dutch and Belgium borders. Five days later German forces crossed into France and, by 20 May, the German armies were heading for the coast. German forces entered Paris on 14 June and a couple of days later France requested armistice terms but fighting continued. The Germans overcame the French at Nancy and, on 20 June, Apéry was captured and became a prisoner-of-war. The armistice was signed on 22 June and within a couple of days all hostilities in France had ended. Apéry, however, was held as a prisoner in Germany where, through the Red Cross, he was able to receive a number of mathematical articles sent by Francesco Severi. Since he was suffering from pleurisy, Apéry was sent back to France for health reasons on 11 June 1941. This was largely due to the efforts of Élie Cartan who had been in contact with the German authorities making a case for his repatriation.

Apéry was discharged on 23 August 1941 and Georges Bruhat, director of the École Normale Supérieure and father of François Bruhat, offered him a research fellowship at the Centre National de la Recherche Scientifique. He was able to undertake research advised by René Garnier, at the same time lecturing at the Sorbonne. He published many papers on Italian style algebraic geometry over the next few years:

*Sur les courbes d'ordre n ayant un point multiple O d'ordre n - 4 et n - 2 tacnodes, les tangentes tacnodales passant par O*Ⓣ (1941) and

*Sur les quintiques à cinq rebroussements*Ⓣ (1941) were followed by four papers in 1942, and another four in 1943. These, of course, were difficult times with Paris under the control of the occupying German forces. Apéry was very active in the French Resistance being director of the Front National, a resistance movement at the École Normale Supérieure, as well as forging identity papers in his room and undertaking other highly dangerous activities. In August 1944 the Gestapo searched the École Normale Supérieure and Apéry, realising what was happening, burnt the incriminating papers he had been forging in his room. Georges Bruhat, director of the École Normale Supérieure, was taken prisoner at this time; he died shortly afterwards in the Buchenwald concentration camp.

Paul Dubreil, who spent the years of World War II in Nancy, returned to Paris in 1946 and advised Apéry on submitting his thesis on algebraic geometry and ideals, which he did in 1947. In the autumn of that year he was appointed as Maître de conferences at Rennes University. Also in 1947 he married Denise Bienaimé; they had three sons, Denys (born 1948), François (born 1950) and Robert (born 1953). François became a mathematician and is the author of [2]. He writes in that article about Apéry's married life:-

His romantic life was troubled. ... a tense and bitter home life ended in divorce in 1971; a second marriage in 1972 was followed by a second divorce in 1977. He could not seem to reconcile family life, mathematical research, and political activism.We note that his second wife was Claudine Lamotte.

Returning to Apéry's career, he was invited to give the prestigious Cours Peccot at the Collège de France in 1948; he spoke on "Algebraic geometry and ideals". He was only on the staff at Rennes for two years before being appointed to the University of Caen in 1949. He remained in Caen for the rest of his career, being promoted to professor in 1953. For many years he continued to have parallel political and mathematical careers. His support of Communist ideas came to an end in 1948 when they supported Trofim Lysenko's ideas about heredity. Lysenko disputed Mendel's theory of heredity claiming that acquired characteristics could be transmitted directly, something which suited the Communist philosophy but was highly unscientific. Apéry became a staunch supporter of Pierre Mendès-France, the radical socialist who was premier of France in 1954. Mendès-France opposed both the Communists and the followers of Charles de Gaulle, and Apéry gave him his full support. In 1958 Apéry became president of the Calvados district of the Parti républicain radical. He continued to take an active role in politics and was sent to Algeria as a reserve lieutenant in December 1959 on a fact finding mission. In May 1968 there was a revolt by students and workers. It rapidly collapsed and, from that time on, Apéry decided to leave politics. In 1969 de Gaulle was defeated and Apéry resigned from the Radical Party feeling that the French Republic was no longer threatened. He retired from his chair at Caen in 1986 but continued to live in Caen. However, his last years before retiring were increasingly difficult ones for he was diagnosed with Parkinson's disease in 1977 and this slowly decreased his mental activities as well as making him increasingly less mobile.

Let us now look at Apéry's mathematical contributions. As we noted above, his early work was on Italian style algebraic geometry. However, in the 1950s he became interested in number theory and worked on diophantine equations. In particular he studied the diophantine equation

$x^{2} + A = p^{n}$, where $A$ and $p$ (a prime) are given.

Two short papers, both entitled *Sur une équation diophantienne*Ⓣ, are devoted to a study of this equation.

He was also interested in the philosophy of mathematics, writing papers such as

*Axiomes et postulats*Ⓣ (1949),

*Le rôle de l'intuition en mathématiques*Ⓣ (1951),

*Les mathématiques sont-elles une théorie pure?*Ⓣ (1952),

*Réforme ou démolition de l'enseignement mathématique*Ⓣ (1971),

*Mathématique constructive*Ⓣ (1982),

*Le temps du mathématicien*Ⓣ (1983), and

*Nature des objets mathématiques*Ⓣ (1986). Pierre Ageron writes [1]:-

Roger Apéry is well known for advocating constructive mathematics and for being a resolute opponent to formalism and Bourbakism. It is less known that he was also one of the first French academics to promote category theory, in spite of its highly structural and formal nature.This mathematical output represents an excellent contribution but in 1977, when he was 61 years old, he produced a result of incredible brilliance. This was his proof that ζ(3) is irrational. Note that ζ(3) is the sum of the reciprocals of the cubes of the natural numbers. Euler had known that

$\zeta(2) = 1/1^{2} + 1/2^{2} + 1/3^{2} + 1/4^{2} + ... = \pi^{2}/6$

and it was also known that $\zeta(2k)$ is irrational but, before Apéry, it was unknown whether $\zeta(2k+1)$ is rational or irrational. Note that it is still an open question whether $\zeta(2k+1)$ is rational for $k > 1$.
For a variety of reasons, mathematicians doubted that Apéry's proof would be correct when it was first announced. The problem had long been an open question, yet Apéry's proof only used methods which had been available for 200 years. Also, very few mathematicians produce their most remarkable piece of work when over 60 years of age. Another possible factor is that it appears that Apéry was not popular with his colleagues. Alfred van der Poorten attended Journées Arithmétiques de Marseille-Luminy in June 1978. He writes [14]:-

The board of programme changes informed us that R Apéry (Caen) would speak Thursday, 14:00 "Sur l'irrationalite de ζ(3)". Though there had been earlier rumours of his claiming a proof, scepticism was general. The lecture tended to strengthen this view to rank disbelief. Those who listened casually, or who were afflicted with being non-Francophone, appeared to hear only a sequence of unlikely assertions. I heard with some incredulity that, for one, Henri Cohen (Grenoble) believed that these claims might well be valid. Very much intrigued, I joined Hendrik Lenstra (Amsterdam) and Cohen in an evening's discussion, in which Cohen explained and demonstrated most of the details of the proof. We came away convinced that Professor Apéry had indeed found a quite miraculous and magnificent demonstration of the irrationality of ζ(3). But we remained unable to prove a critical step. ... We were quite unable to prove that [certain sequences satisfied a recurrence as claimed by Apéry]. (Apéry rather tartly pointed out to me in Helsinki that he regarded this more a compliment than a criticism of his method). But empirically (numerically) the evidence in favour was utterly compelling. It seemed indeed that ζ(3) had been proved irrational ...Philip Gibbs gives a more colourful account of Apéry's conference announcement in [11]:-

Neither Cohen nor I had been able to prove [the missing steps] in the intervening two months [before we attended the International Congress of Mathematicians at Helsinki in August 1978]. After a few days of fruitless effort the specific problem was mentioned to Don Zagier (Bonn), and with irritating speed he showed that indeed the sequence satisfies the recurrence. This more or less broke the dam and [the missing steps] were quickly conquered. Henri Cohen addressed a very well-attended meeting at 17:00 on Friday, August 18 in the language of the majority, proving [the missing step] and explaining how this implied the irrationality of ζ(3). Apéry then made some remarks on the status of the French language, and alluded to the underlying motivation for his astonishing proof. ... It is some measure of Apéry's achievement that these questions have been considered by mathematicians of the top rank over the past few centuries without much success being achieved. ... Apéry's incredible proof appears to be a mixture of miracles and mysteries. ... Most startling of all though should be the fact that Apéry's proof has no aspect that would not have been accessible to a mathematician of 200 years ago. The proof we have seen is one that many mathematicians could have found, but missed.

In 1978 he presented a lecture on his proof at the Journées Arithmétiques de Marseille which was greeted with doubt and disbelief. Each step he wrote on the blackboard appeared to be a remarkable identity that his audience considered unlikely to be true. When someone asked him "where do these identities come from?" he replied "They grow in my garden." Obviously this did not boost anyone's confidence. Nevertheless, a few mathematicians recognised that there was something significant in the proposed proof. They checked the identities numerically and found that they did indeed seem to hold. It was not long before the full validity of Apéry's work was confirmed and the sceptics were forced to eat their words.In fact Frits Beukers managed to produce a simpler version of Apéry's remarkable proof which he published in [9]. Beukers writes in [8]:-

It has been my good fortune to find a very simple version of the proof a few months after Apéry's announcement.Apéry was honoured by being made a Knight of the Légion d'Honneur in December 1970. He eventually succumbed to Parkinson's disease in 1994 after years of ill health. He was buried in Paris, with his parents, in a small tomb set in a wall of Père Lachaise cemetery's Columbarium. A small plaque contains the names of Apéry and his parents, and the formula:

$\large\frac{1}{1}\normalsize + \large\frac{1}{8}\normalsize + \large\frac{1}{27}\normalsize + \large\frac{1}{64}\normalsize + ... ≠ \large\frac{p}{q}\normalsize .$

### References (show)

- P Ageron, La philosophie mathématique de Roger Apéry,
*Philosophia Scientiae*, cahier spécial**5**(2005), 233-256. - F Apéry, Roger Apéry, 1916-1994: A Radical Mathematician,
*The Mathematical Intelligencer***18**(2) (1996), 54-61. - R Apéry, Mathématique constructive,
*Collect. Points Sér. Sci.***29**(Seuil, Paris, 1982), 58-72. - R Apéry, Les mathématiques sont-elles une théorie pure?,
*Dialectica***6**(1952), 309-310. - R Apéry, Le rôle de l'intuition en mathématiques, in
*Congrès International de Philosophie des Sciences, Paris, 1949***III**(Hermann & Cie., Paris, 1951), 85-88. - R Apéry, Axiomes et postulats, in
*Library of the Tenth International Congress of Philosophy, Amsterdam, 11-18 August 1948***I**(1949), 708-710. - T M Apostol, A proof that Euler missed: evaluating ζ(2) the easy way,
*The Mathematical Intelligencer***5**(3) (1983), 59-60. - F Beukers, Consequences of Apéry's work on ζ(3) (2003).
- F Beukers, A note on the irrationality of ζ(2) and ζ(3),
*Bull. London Math. Soc.***11**(1979), 268-272. - M Black, Review: Axiomes et postulats by Roger Apéry,
*The Journal of Symbolic Logic***14**(3) (1949), 183. - P E Gibbs, 'Crackpots' Who Were Right II,
*Prespacetime Journal***1**(3) (2010), 489-497. - Y Hellegouarch, Roger Apéry (1916-1994) (French),
*Gaz. Math. No.***64**(1995), 82-83. - M Mendès-France, Roger Apéry et l'irrationnel,
*La Recherche***97**(February 1979), 170-172. - A van der Poorten, A proof that Euler missed ...,
*The Mathematical Intelligencer***1**(4) (1979), 195-203.

### Additional Resources (show)

Other websites about Roger Apéry:

Written by J J O'Connor and E F Robertson

Last Update January 2013

Last Update January 2013