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# Roger Apéry (1916-1994) © 2003-2016   Gérard P. Michon, Ph.D.

The ashes of Roger Apéry are stored with those of his parents in columbarium number 7972 at the Père Lachaise cemetery in Paris (France) behind a plaque where his most famous result is engraved:

1 + 1/8 + 1/27 + 1/64 + ...   ¹   p/q

This expresses the irrationality of z(3), the sum of reciprocal cubes  ("Apéry's Constant").  Roger Apéry earned international fame in 1977, near the end of his career, when he came up with a miraculous proof of this, using "Apéry's numbers"  (A005259).

Apéry's result remained superbly isolated until 2000, when Tanguy Rivoal showed that infinitely many values of the zeta function at odd integers are irrational.  In 2001, Wadim Zudilin proved that at least one of the 4 values z(5), z(7), z(9) and z(11) is irrational.  Everybody's guess is that all such values are transcendental, but it's only a guess at this point...  By tantalizing contrast, Euler showed, in 1735, that the value z(2n) at an  even  integer  2n  is a rational multiple of the 2n-th power of p, starting with  z(2) = p2/6  (the former Basel Problem)  and  z(4) = p4/90 .

### Roger Apéry, French mathematician (1916-1994)

The biographical data about Roger Apéry is found in texts by his son, the mathematician  François Apéry (b. 1950)  under titles alluding to his father's "radical" political affiliation.  (The term refers to a French party, which Apéry joined in 1934; the literal translation is misleading.)

• Roger Apéry, 1916-1994:  A Radical Mathematician  by François Apéry.
The Mathematical Intelligencer, vol. 18, 2  (1996)  pages 54-61.
• French translation of the above by Pierre Karila and Mireille Saunier.
Notes pour la famille de Roger Apéry (1916-1994).
This 176-page text by F. Apéry was privately printed in september 1998.  We thank for the use of her copy Jeanne Refleu, widow of Lucien Refleu, a lifelong friend and colleague of Roger Apéry's in Caen (Normandy).

### Origins :

Roger Apéry was born in Rouen (Normandy, France) on November 14, 1916.  He was the son of:
• Georges Apéry (1887-1978).  A Greek national born in Constantinople (Istanbul) who emmigrated to France in 1903 to study engineering, he was admitted to the  Institut Electrotechnique de Grenoble  (now, "ENSI Grenoble").  Georges voluntereed into the French army in 1914 to acquire French citizenship.  He fought in the Dardanelles in 1915, got typhoid fever and came back on a hospital ship.
• Justine Van Der Cruyssen (1892-1965).  An occasional piano teacher, she was known as Lise or, later, Louise.  She also gallicized her Flemish maiden name to Delacroix...  Louise  passed on her musical skills to her son.

Roger's parents were married in Rouen, when Georges was on military leave.  They lived in Lille from 1920 to 1926, then established themselves in Paris:  First in a slum at 58, rue de Paradis (Paris X) then in a small apartment on 52, rue de la Goutte d'Or (Barbès, Paris XVIII) where they would live extremely modestly from 1927 on...  There was no room for their only child, so Roger had to go to boarding school.  Like his father before him, Roger Apéry saw education and academic success as the only way out of his original condition.  This remained the focal point of his value system throughout his life...

### Some Milestones in the Life of Roger Apéry :

• Born in Rouen, on November 14, 1916.
• Concours Général laureate (3rd junior prize in maths) in 1932.
• Rewarded again in maths (2nd senior prize) & Physics in 1933.
• April 1934:  Joins the new "radical-socialiste" party of Camille Pelletan.
• In spite of a 19/20 in maths, fails to enter the "rue d'Ulm" school in 1935.
• Enters the prestigious ENS (rue d'Ulm) in 1936, ranked second nationwide.
• Bracketed first (with Jacqueline Lelong-Ferrand) at the agrégation in 1939.
• Drafted on September 16, 1939.
• Junior lieutenant in Nancy (145th Artillery) in February 1940.
• Prisoner of war, captured in Nancy on June 20, 1940.
• Released for health reasons on June 11, 1941.  Discharged on August 23.
• Fall 1941:  Assistant lecturer at the Sorbonne (sponsored by Elie Cartan).
• Doctorate awarded in 1947 (thesis under Paul Dubreil & René Garnier).
• Fall 1947:  Youngest Maître de conferences in France (Rennes University).
• Married to Denise Bienaimé from 1947 to 1971 (three sons).
• 1948: Birth of eldest son, Denys Apéry.
• At the University of Caen from 1949 to 1986 (named professor in 1953).
• 1950: Birth of second son, François Apéry.
• 1953: Birth of third son, Robert Apéry.
• President for the Calvados district of the Parti républicain radical, in 1958.
• Serves as a reserve lieutenant in Algeria, in December 1959.
• 1966:  Reactivates the "Cercle Alekhine" chess club in Caen.
• Chevalier de la Légion d'Honneur, in December 1970.
• Married to Claudine Lamotte from 1972 to 1977.
• 1977: Proves the irrationality of the constant z(3), now named after him.
• Retires in 1986.
• Dies on December 18, 1994.

Apéry never went for  Bourbakism.  He adopted  Category theory  early on.

## Mathematical Anecdotes

### "Bombieri's Napkin Problem" :  A Botched Challenge (1979)

During the 1979  Queen's Number Theory Conference in Kingston (Ontario), Enrico Bombieri (Fields Medalist in 1974) offered  in jest  a challenge analoguous to Fermat's Last Theorem to some colleagues he was having diner with, including  Roger Apéry  and  Michel Mendès-France  (who reported the anecdote).

Prove that there are no nontrivial solutions, in positive integers,
to the following equation
(involving choice numbers):

C(x,n) + C(y,n)   =   C(z,n)

The next morning, Apéry offered a solution:  n = 3,  x = 10,  y = 16,  z = 17. Bombieri just replied, with a straight face:  "I said nontrivial."

The puzzle is much older than this anecdote:  It is casually mentioned in the popular book  Tomorrow's Math: Unsolved Problems for the Amateur  (1972 edition, at least)  by  Dr. Charles Stanley Ogilvy  (1913-2000)  who echoes a misleading presentation that could easily have fooled the likes of Bombieri or Mendès-France in an era when computer access wasn't easy...
2006-01-30:  Some Solutions to
C(x,n) + C(y,n)   =   C(z,n)
nxyz
n 2n-12n-12n
1 xyx+y
2 3m+34m+35m+4
3m+44m+65m+7
5m+512m+1013m+11
5m+612m+1513m+16
15m+108m+517m+11
15m+218m+1217m+24
x½ x(x-1)½ x(x-1) + 1
3 101617
225657
325760
417276
63104111
8692112
36120121
4 132190200
6 141516
191921
35 118118120
40 103104105

With y=x+i and z=x+j (i<j) we may factor out C(x,n-j) to obtain an equation of degree j in x and n.  The second line of the above table (n=1) corresponds to j=1.

For j=2,  we obtain two quadratic diophantine equations  (i=0 or i=1)  respectively yielding the following two infinite families of explicit solutions:

1)   When  8n2+1  is a perfect square  (q2 )  a solution is:  x = ½ (4n-3+q),  y = x, z = x+2  Such values of  n  (and those of q)  obey the recurrence:  ai+2 =  6 ai+1 - ai

  n  x y z 1 6 35 204 1189 6930 40391 235416 7997214 2 19 118 695 4058 23659 137902 803759 27304195 2 19 118 695 4058 23659 137902 803759 27304195 4 21 120 697 4060 23661 137904 803761 27304197

2)   If  5n2-2n+1  is a square  (q2 )  then a solution is:  x = ½ (3n-3+q),  y = x+1, z = x+2  Such values of  n  obey the recurrence:  ai+2 =  7 ai+1 - ai - 1
(Every fourth Fibonacci number is a value of q :   2, 13, 89, 610, 4181, 28657...)

  n  x y z 1 6 40 273 1870 12816 87841 602070 4126648 1 14 103 713 4894 33551 229969 1576238 10803703 2 15 104 714 4895 33552 229970 1576239 10803704 3 16 105 715 4896 33553 229971 1576240 10803705

Bombieri's Napkin Problem is now discussed on the "Math Overflow" forum :-)   For undisclosed reasons,
Wadim Zudilin states for the record that he "personally doesn't like" the above presentation :-(

## Obituary

[NA Digest] From: Marc Prévost   prevost@lma.univ-littoral.fr
Date: Wed, 21 Jun 1995 08:48:35 --100
Subject: Roger Apéry

Professor Roger Apéry, a prominent figure of the University of Caen (France), passed away after a prolonged illness on [the 18th of] December 1994. Roger Apéry was born in Rouen in 1916 of a mother of Flemish origin and a Greek father who had volunteered to serve in the French army in 1914 in order to obtain French nationality.

After brilliant studies at the "Lycée Louis Le Grand" where he distinguished himself several times in the "Concours général", Roger Apéry was placed second among entrants at the ENS in 1936 and came first at the "Agrégation de Mathématiques".

Called up for the army in 1939 and a prisoner of war in June 1940, he was released in October 1941 for health reasons.  Appointed assistant lecturer at the Sorbonne in 1942, he joined a group of ENS students in the French Résistance and became the leader of the National Front at the ENS.  In 1947 he defended his thesis in algebraic geometry "à l'italienne" under the supervision of Paul Dubreil and was appointed Lecturer at Rennes (the youngest ever in France).  From 1949 until he retired in 1986, Roger Apéry was a Professor at the University of Caen where he created a research team on algebra and number theory.

At the end of his career, in 1977, he made a sensational discovery which was to make his name famous throughout the world.  His proof of the irrationality of the sum of the inverse of the cubes of integers by an exceptionally clever method worthier of his Greek ancestors than of Bourbaki, made him a legend.  In addition to a keen sense of provocation, Roger Apéry enjoyed playing the piano --his mother had taught him-- chess, philosophy and...  politics.  Having joined the Camille Pelletan Radical Party at a young age after the riots of 1934, he resigned after Munich.  Then, at the end of the war, he once again became an active party member with Pierre Mendès-France.  As president of the Calvados Radical Party in the 60's he remained active in politics until May 68.  Being opposed to the reforms instituted after 68 by Edgar Faure, he abandoned political life when he realized University life was running against the tradition he had always upheld.

Many researchers have worked with the so-called Apéry sequences to

• rediscover his proof of the irrationality of z(3)
(H. Cohen, A. Van Den Poorten, E. Reyssat, F. Beukers, M. Prevost)
• generalize his recurrence relation in connexion with Numerical Analysis and Orthogonal Polynomials   (R. Askey, J.A. Wilson, A.L. Schmidt)
• study the congruence properties of Apéry numbers
( P.T. Young, Y. Mimura, I. Gessel, S. Chowla)

[From a text by Y. Hellegouarch.]

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