# Édouard Zeckendorf

### Quick Info

Born
2 May 1901
Liège, Belgium
Died
16 May 1983
Liège, Belgium

Summary
Edouard Zeckendorf was a Belgian medical doctor and mathematician best known for his work on Fibonacci numbers.

### Biography

Édouard Zeckendorf's parents were Abraham Zeckendorf and Henriette van Gelder. Dr Abraham Zeckendorf was a dentist and he set up a dental practice in Liège near the end of the 19th century. The family were Dutch and Abraham Zeckendorf was a practicing Jew. Abraham and Henriette's son Eduourd was born and brought up in Liège. He learnt both Dutch, the language his parents spoke, and French, the official language of Liège. Eduourd entered the Royal Athenaeum of Liège in 1912. There he studied the classical languages of Greek and Latin, and the modern languages of English and German. He also studied mathematics and drawing, two subjects in which he was particularly talented and which he greatly loved. Of course, Eduourd's school years were difficult ones due to World War I. The German army attacked Liège in August 1914, the first military action in the war, and the city surrendered on 7 August. The city was occupied by Germany for the whole of the war so almost all of Zeckendorf's school education took place under German occupation. The war ended in 1918 and Zeckendorf graduated from the school in the following year following which he began his studies at the University of Liège.

Zeckendorf had decided that, like his father, he would follow a medical profession. At the University of Liège he studied to become a medical doctor, and in 1925 he qualified. He had specialised in surgery and, once he qualified, he became an officer in the medical corps of the Belgium army. Following even more closely his father's profession, Zeckendorf worked towards a Dental Surgery License between 1927 and 1931. Zeckendorf married Elise Schwers (born Liège on 2 July 1889), a nurse known as Elsa, on 29 June 1929. Elsa's parents were Jean Henri Schwers (1853-1923) and Anna Ewaldine Wesche (1858-1939). Eduourd and Elsa Zeckendorf had a major common interest. Both were artists and, before her marriage, Elsa had made many fine drawings of Paris scenes, some in pencil and some in charcoal. She also painted with oils and continued with this after her marriage. Zeckendorf too was keen on drawing and this was a hobby which he enjoyed whenever he had some spare time. Together they attended art exhibitions and made friends with the best artists in and around Liège. Shortly after he married, by 1930, Zeckendorf was in charge of the Saint Laurent Hospital in Liège, a military hospital run by the Belgium army.

Zeckendorf had another hobby, in addition to art, for he was an excellent amateur mathematician. He has given his name is given to the property that every positive integer can be represented uniquely as the sum of non-consecutive Fibonacci numbers, the sequence defined by
$F_{1} = F_{2} = 1$ and $F_{n} = F_{n-1} + F_{n-2}$ for $n > 2$.
This is called Zeckendorf's theorem, and the subsequence of Fibonacci numbers which add up to a given integer is called its Zeckendorf representation. (Because $F_{1} = F_{2}$, we need to exclude $F_{1}$ from the representation to give uniqueness.) For example,
71 = 55 + 13 + 3,
1111 = 987 + 89 + 34 + 1.
Zeckendorf had obtained these results, and other results concerning sequences, by 1939 but before he was able to publish them World War II broke out. At first German troops invaded Poland and then in April 1940 Germany invaded Denmark and Norway. Only in May 1940 did Germany push west when, on 10 May, they invaded Holland, Belgium and Luxembourg. Zeckendorf's parents fled as soon as the invasion began and they settled in Nice, France. By 14 May German troops had reached Liège where Zeckendorf continued to work at the hospital. An attempt by British and French troops to counterattack at Arras on 21 May failed and, on 28 May, King Leopold of Belgium signed an armistice with the Germans. At this point the Belgium army ceased to exist and Zeckendorf was taken prisoner and interned as a prisoner of war until 1945 [1]:-
As a prisoner of war, he stayed in several oflags until 1945. (An 'Oflag' was an 'Officierenlager', a camp for imprisoned officers, as contrasted to a 'Stalag'.) During his captivity Dr Zeckendorf provided medical care to allied prisoners of war. He also sketched soldiers representing the many various peoples of the Soviet Union. Zeckendorf escaped from a camp, and afterwards, his status as a nonpracticing Jew was ignored by the Germans. ... Zeckendorf chose to continue his care of prisoners of war in Germany despite opportunities to return to his home.
Elsa Zeckendorf died suddenly in 1944 and, a few months later, Zeckendorf's father Abraham died in Nice. In April 1945 allied troops rapidly pushed the German armies back and prisoners were liberated from the camps. Zeckendorf was liberated and, after the surrender of all German troops in Belgium in early May, he was able to return to his home in Liège. He found that his home was occupied by American troops. The Zeckendorf family home had been classed as "abandoned" by the Germans following the Belgium surrender in May 1940 and the house had been occupied by German troops who destroyed most of the contents. Following the German surrender in 1945, American troops moved into the house and that was the state of affairs when Zeckendorf reached Liège. His only close relation who remained alive was his mother, who was still living in Nice, and he decided to go to Nice to look after his mother who, by this time, was old and in poor health. For his service during World War II, Zeckendorf received numerous awards: Officer of the Order of the Oak Crown (1946) (Officer is the second highest of five classes of the Order of the Oak Crown); Prisoner of War Medal (1946); Officer of the Order of Leopold (1949); and Officer of the Order of Leopold II (1950).

In August 1947 Pakistan had become an independent country. However, there were great tensions during the following year regarding the border between Pakistan and India. The United Nations sent a Commission in January 1949 to end the localised warfare which had gone on during 1948. On 16 March 1949 Zeckendorf went to India at the head of a Belgium contingent which was part of the United Nations effort. He remained there for a year, during which time (July 1949) a ceasefire was agreed between India and Pakistan. This resulted in a 800 km ceasefire line between the two countries and Zeckendorf was put in charge of inspecting this ceasefire line. It was during this period in his life that his first mathematical publications appeared: Étude fibonaccienne. Arrangements avec répétition de lettres a et de chaînes limitées de lettres b (1949); Étude fibonaccienne. De certaines coupes obliques parallèles dans les polytopes arithmétiques à $(p - 1)$ dimensions (1950); and Étude fibonaccienne (1951). These three papers were all published in Mathesis. He subsequently published over 20 further mathematical papers, nearly all of them in the Bulletin de la Société Royale des Sciences de Liège, mainly on elementary number theory. MathSciNet list 53 papers with Zeckendorf's name in the title. It occurs as "Zeckendorf numbers", "Zeckendorf representations", "Zeckendorf decompositions", "Zeckendorf trees", "Zeckendorf arrays", "Zeckendorf identities", and "Zeckendorf expansions". There are over 100 papers in which Zeckendorf's name appears in the review.

Zeckendorf was elected as an associate member of the Societe Royale des Sciences de Liège on 20 June 1957. Just over two years later, he married for the second time [1]:-
Zeckendorf married Marie Jeanne Lempereur in Brussels, Belgium, on 27 July 1959. Miss Lempereur's family was Belgian but had lived in Manitoba, Canada, at the time of her birth in 1908. When she was a young girl, the family had returned to Belgium. During the eighteen years of their marriage, Dr and Mrs Zeckendorf enjoyed an active life, visiting exhibits and museums, travelling and visiting cities of artistic interest, and reading. After his second wife's death in July 1977, Dr Zeckendorf continued his activities, even after the discovery of cancer. Near the end of his life, he often visited friends in Liège, and he regularly attended the monthly meetings of the Societe Royale des Sciences de Liège.

### References (show)

1. C Kimberling, Edouard Zeckendorf, Fibonacci Quart. 36 (5) (1998), 416-418.
2. C Kimberling, Corrigendum: 'Edouard Zeckendorf', Fibonacci Quart. 37 (1) (1999), 33.
3. G M Phillips, Edouard Zeckendorf, Personal Communication (August 2000).
4. G M Phillips, Zeckendorf representation, Encyclopaedia of Mathematics, Supplement II (Kluwer, 2000), 525-526.
5. E Zeckendorf, Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41 (1972), 179-182.
6. E Zeckendorf, A generalized Fibonacci numeration, Fibonacci Quart. 10 (4) (1972), 365-372.

### Additional Resources (show)

Other websites about Édouard Zeckendorf:

Written by J J O'Connor, E F Robertson and G M Phillips, St Andrews
Last Update July 2012