Lothar Collatz

Quick Info

Born

6 July 1910
Arnsberg, Westphalia, Germany

Died

26 September 1990
Varna, Bulgaria

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Summary

Lothar Collatz was a German mathematician who worked in numerical analysis and is best known for the eponymous "3x + 1 problem".

Biography

Lothar Collatz, like most German students of his time, studied at a number of different universities. He entered the University of Greifswald in 1928, moving to Munich, then to Göttingen, and finally to Berlin where he studied for his doctorate under Alfred Klose. Collatz was awarded his doctorate in 1935 for his dissertation Das Differenzenverfahren mit höherer Approximation für lineare Differentialgleichunge Ⓣ. In [8] Meinardus and Nürnberger write:-

He often told how much he had been impressed by the lectures of Hilbert, Courant, von Mises, Schur, and other famous mathematicians of that period. He was convinced that mathematics and mathematicians had a responsibility to apply their results to, and be motivated by, real world phenomena. he never wearied of fighting for this conviction.

Many will know the name of Collatz today because of the "Collatz problem". In many ways it might seem a pity that a mathematician who has produced so much important and fundamental work should be most remembered for a novelty, yet this problem has intrigued mathematicians ever since he proposed it in 1937. The Collatz problem is simple to state. Define a function $f$ on the positive integers by

$f (n) = 3n+1$ if $n$ is odd;
$f (n) = \Large\frac n 2$ if $n$ is even.

Given any integer $m$ define a sequence by putting

$a(1) = m$ and, for $i ≥ 1, a(i+1) = f (a(i))$.

The problem asks if, for every starting value $m$, the sequence $a(i)$ always reaches 1? The problem remains unsolved, but before you try a few small numbers yourself looking for a counterexample, let us say that the conjecture has been verified for all numbers $m$ up to about $10^{14}$.

In 1943 Collatz was appointed to a professorship at the Technical University of Hanover. After holding this position for nine years, in 1952 he moved to the University of Hamburg. There he founded the Institute of Applied Mathematics.

MathSciNet lists 238 items under Collatz' name. Most of these publications are in numerical analysis but those range through almost every area within the subject. Among his early papers are Genäherte Berechnung von Eigenwerten Ⓣ (1939) in which he considers various methods of approximating characteristic values, Das Hornersche Schema bei komplexen Wurzeln algebraischer Gleichungen Ⓣ (1940) in which he presents a more efficient way of using Horner's method to approximate the complex roots of an algebraic equation, and Schrittweise Näherungen bei Integralgleichungen und Eigenwertschranken Ⓣ (1940) in which inequalities between the eigenvalues of certain integral equations are studied.

Collatz began publishing important books early in his career. Eigenwertprobleme und ihre numerische Behandlung Ⓣ (1945) contains three parts, the first containing a collection of practical applications which lead to boundary value problems for ordinary and partial differential equations. The second part provides the theory of boundary value problems while the final part provides numerical method to solve these. This was followed by Numerische Behandlung von Differentialgleichungen Ⓣ (1951) which provides a comprehensive text on numerical methods for solving differential equations. Collatz was keen to provide methods for scientists as in Differentialgleichungen für Ingenieure: Eine Einführung Ⓣ (1960):-

This small book gives a wealth of information on differential equations. ... The concise formulation is clear and precise, and it will appeal to engineers on account of its many graphic examples.

An interesting combining of two areas was presented in Functional analysis and numerical mathematics (1966), which was an English translation of a German book published two years earlier. A S Householder writes:-

It seems strange that this book should be the first of its kind, since it hardly needs to be said that "numerical mathematics" must draw heavily from functional analysis. Nevertheless, in an unnecessarily modest preface, the author disclaims any intention of writing a textbook on either functional analysis or numerical mathematics, offering the book instead as illustrating the artificiality of any separation of applied from pure mathematics. In spite of the disclaimer, the book could be used as a basis for a rather extensive course on functional analysis for numerical analysts, and the footnotes and the bibliography provide for a considerable amount of collateral reading.

A very different area was covered in Optimierungsaufgaben Ⓣ (1966) (written jointly with W Wetterling). This is an introductory textbook which introduces students to linear programming, two-person-zero-sum games, quadratic programming, and convex programming. The book Aufgaben aus der Angewandten Mathematik Ⓣ (1972) (with J Albrecht) provides a collection of problems (with their solutions) on the solution of equations and systems of equations, interpolation, quadrature, approximation, and harmonic analysis. Approximationstheorie. Tschebyscheffsche Approximation mit Anwendungen Ⓣ (1973) (with W Krabs) is described by J L Ullman as follows:-

[T]he total effect is to provide a stimulating introduction to the subject to people with both pure and applied inclinations, and at the same time, providing a good primary or secondary reference for an advanced undergraduate course, or a beginning research seminar in approximation theory and/or numerical analysis.

Later texts by Collatz include Optimization problems (1975) and Differential equations (1986), the second of these being an English translation of an earlier German book.

Collatz received many honours for his contributions including election to the German Academy of Scientists Leopoldina, the academy at Bologna and that at Modena. He was made an honorary member of the Hamburg Mathematical Society and given honorary degrees by the University of São Paulo, the Technical University of Vienna, the University of Dundee in Scotland, Brunel University in England, the Technical University of Hanover, and the Technical University of Dresden.

Meinardus and Nürnberger write [6]:-

Professor Collatz was a truly wonderful individual. He was modest in his behaviour, and ever amiable and helpful.

He died in Varna, Bulgaria, while attending a mathematics conference on Computer Arithmetic.

References (show)

1. J Albrecht, P Hagedorn and W Velte, Lothar Collatz (German), Numerical treatment of eigenvalue problems, Vol. 5, Oberwolfach, 1990 (Birkhäuser, Basel, 1991), viii-ix.
2. R Ansorge, Lothar Collatz (6 July 1910-26 September 1990) (German), Mitt. Ges. Angew. Math. Mech. No. 1 (1991), 4-9.
3. U Eckhardt, Der Einfluss von Lothar Collatz auf die angewandte Mathematik, Numerical mathematics, Sympos., Inst. Appl. Math., Univ. Hamburg, Hamburg, 1979 (Birkhäuser, Basel-Boston, Mass., 1979), 9-23.
4. L Elsner and K P Hadeler, Lothar Collatz - on the occasion of his 75th birthday, Linear Algebra Appl. 68 (1985), vi; 1-8.
5. R B Guenther, Obituary : Lothar Collatz, 1910-1990, Aequationes Math. 43 (2-3) (1992), 117-119.
6. H Heinrich, Zum siebzigsten Geburtstag von Lothar Collatz, Z. Angew. Math. Mech. 60 (5) (1980), 274-275.
7. G Meinardus, G Nürnberger, Th Riessinger and G Walz, In memoriam : the work of Lothar Collatz in approximation theory, J. Approx. Theory 67 (2) (1991), 119-128.
8. G Meinardus and G Nürnberger, In memoriam : Lothar Collatz (July 6, 1910-September 26, 1990), J. Approx. Theory 65 (1) (1991), i; 1-2.
9. J R Whiteman, In memoriam : Lothar Collatz, Internat. J. Numer. Methods Engrg. 31 (8) (1991), 1475-1476.

Additional Resources (show)

Other websites about Lothar Collatz:

1. Mathematical Genealogy Project
2. MathSciNet Author profile
3. zbMATH entry

Cross-references (show)

1. Other: Earliest Known Uses of Some of the Words of Mathematics (R)

Written by J J O'Connor and E F Robertson
Last Update November 2006