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... Leonhard Euler (1707-1783) 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 .
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'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...
Apéry never went for Bourbakism. He adopted Category theory early on.
During the 1979 Queen's Number Theory Conference in Kingston (Ontario), Enrico Bombieri (Fields Medalist in 1974) offered in jest a challenge analogous 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." Just a joke!
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年01月30日: Some Solutions to
C(103.40) = 61218182743304701891431482520 is one of the rare numbers found 5 times or more in Pascal's triangle (A003015). Likewise for C(14,6) = 3003. See entertaining video by Zoe Griffiths (9:05, 2020年04月19日).
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
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...)
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 :-(
[NA Digest]
From: Marc Prévost
prevost@lma.univ-littoral.fr
Date: 1995年6月21日 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
[From a text by Y. Hellegouarch.]
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