Derrick Henry Lehmer
Encyclopedia
Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991) was an American mathematician who refined Édouard Lucas
' work in the 1930s and devised the Lucas–Lehmer test
for Mersenne prime
s. Lehmer's peripatetic career as a number theorist
, with he and his wife taking numerous types of work in the United States and abroad to support themselves during the Great Depression
, fortuitously brought him into the center of research into early electronic computing.
, to Derrick Norman Lehmer
, a professor of mathematics
at the University of California, Berkeley
, and Clara Eunice Mitchell.
He studied physics
and earned a Bachelor degree from UC Berkeley, and continued with graduate studies at the University of Chicago
.
He and his father worked together on Lehmer sieve
s.
(November 6, 1906-May 7, 2007), a Russian student of his father's, who had begun with work toward an engineering
degree but had subsequently switched focus to mathematics, earning her B.A. in 1928. Later that same year, Lehmer married Emma and, following a tour of Northern California and a trip to Japan to meet Emma's family, they moved by car to Providence, Rhode Island
, after Brown University
offered him an instructorship.
and a Ph.D.
, both from Brown University, in 1929 and 1930, respectively; his wife obtained a Master's degree in 1930 as well, coaching mathematics to supplement the family income, while also helping her husband type his Ph.D. thesis, An Extended Theory of Lucas' Functions, which he wrote under Jacob Tamarkin
.
from 1930 to 1931 and at Stanford University
from 1931 to 1932. In the latter year, the couple's first child Laura was born.
After being awarded a second National Research Fellowship, the Lehmers moved on to Princeton, New Jersey
between 1932 and 1934, where Dick spent a short time at the Institute for Advanced Study
.
He worked at Lehigh University
in Pennsylvania
from 1934 until 1938. Their son Donald was born in 1934 while Dick and Emma were at Lehigh.
The year 1938-1939 was spent in England
on a Guggenheim Fellowship
visiting both the University of Cambridge
and the University of Manchester
, meeting G. H. Hardy
, John Edensor Littlewood
, Harold Davenport
, Kurt Mahler
, Louis Mordell
, and Paul Erdős
. The Lehmers returned to America by ship with second child Donald just before the beginning of the Battle of the Atlantic.
Lehmer continued at Lehigh University for the 1939-1940 academic year.
(pseudorandom number generator
), which is frequently referred to as a Lehmer random number generator. The Lehmers also assisted Harry Vandiver
with his work on Fermat's Last Theorem
, computing many Bernoulli numbers required.
Lehmer was chairman of the Department of Mathematics at University of California, Berkeley
from 1954 until 1957. He continued working at UC Berkeley until 1972, the year he became professor emeritus.
, a group established as part of the Ballistics Research Laboratory to prepare the ENIAC
for utilization following its completion at the University of Pennsylvania
's Moore School of Electrical Engineering
; the other Computations Committee members were Haskell Curry
, Leland Cunningham
, and Franz Alt
. It was during this short tenure that the Lehmers ran some of the first test programs on the ENIAC—according to their academic interests, these tests involved number theory, especially sieve methods
, but also pseudorandom number generation. When they could arrange child care, the Lehmers spent weekends staying up all night running such problems, the first over the Thanksgiving
weekend of 1945. (Such tests were run without cost, since the ENIAC would have been left powered on anyway in the interest of minimizing vacuum tube failures.) The problem run during the 3-day Independence Day
weekend of July 4, 1946, with John Mauchly
serving as computer operator, ran around the clock without interruption or failure. The following Tuesday, July 9, 1946, Lehmer delivered the talk "Computing Machines for Pure Mathematics" as part of the Moore School Lectures
, in which he introduced computing as an experimental science, and demonstrated the wit and humor typical of his teaching lectures.
Lehmer would remain active in computing developments for the remainder of his career. Upon his return to Berkeley, he made plans for building the California Digital Computer (CALDIC
) with Paul Morton and Leland Cunningham.
, a policy initiated by the Board of Regents of the State of California in 1950 during the Communist scare personified by Senator Joseph McCarthy
. Lehmer took a post as Director of the National Bureau of Standards' Institute for Numerical Analysis (INA), working with the Standards Western Automatic Computer (SWAC
). On October 17, 1952, the State Supreme Court proclaimed the oath unconstitutional, and Lehmer returned to Berkeley shortly thereafter.
Lehmer had quite a wit. On the occasion of the first Asilomar number theory conference, which became an annual event, Lehmer, as the organizer, was inspecting the facilities of the Asilomar Conference Grounds
--basically a wooden building on the beach. Someone said they couldn't find a blackboard and Lehmer spotted some curtains in the middle of the wall. Moving the curtains aside revealed a very small blackboard, whereupon Lehmer said "Well, I guess we won't be doing any analytic number theory!"
and participated in the Cunningham project
.
, known mainly as a pioneer in number theory computing, also made major contributions to combinatorial computing, having devised algorithms for efficiently generating all the permutations on n elements He also developed two algorithms, rank(p) and unrank(k). Given a permutation p, if k = rank(p), then p is the kth permutation. Given an integer k, unrank(k) is the kth permutation. D N Lehmer's method uses his so called factorial representation of the integer k. See section 5.1 in Permutation
.
D. H. Lehmer continued his fathers interest in combinatorial computing and in fact wrote the article "Machine tools of Computation," which is chapter one in the book "Applied Combinatorial Mathematics," by Edwin Beckenbach, 1964. It describes methods for producing permutations, combinations etc. This was a uniquely valuable resource and has only been rivaled recently by Volume 4 of Donald Knuth's series.
Edouard Lucas
François Édouard Anatole Lucas was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.-Biography:...
' work in the 1930s and devised the Lucas–Lehmer test
Lucas–Lehmer test
In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known. It is the basis of the Pratt certificate that gives a concise verification that n is prime....
for Mersenne prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
s. Lehmer's peripatetic career as a number theorist
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, with he and his wife taking numerous types of work in the United States and abroad to support themselves during the Great Depression
Great Depression
The Great Depression was a severe worldwide economic depression in the decade preceding World War II. The timing of the Great Depression varied across nations, but in most countries it started in about 1929 and lasted until the late 1930s or early 1940s...
, fortuitously brought him into the center of research into early electronic computing.
Early life
Lehmer was born in Berkeley, CaliforniaBerkeley, California
Berkeley is a city on the east shore of the San Francisco Bay in Northern California, United States. Its neighbors to the south are the cities of Oakland and Emeryville. To the north is the city of Albany and the unincorporated community of Kensington...
, to Derrick Norman Lehmer
Derrick Norman Lehmer
Derrick Norman Lehmer was an American mathematician and number theorist....
, a professor of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
at the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
, and Clara Eunice Mitchell.
He studied physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and earned a Bachelor degree from UC Berkeley, and continued with graduate studies at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
.
He and his father worked together on Lehmer sieve
Lehmer sieve
Lehmer sieves are mechanical devices that implement sieves in number theory. Lehmer sieves are named for Derrick Norman Lehmer and his son Derrick Henry Lehmer...
s.
Marriage
During his studies at Berkeley, Lehmer met Emma Markovna TrotskaiaEmma Lehmer
Emma Markovna Lehmer was a mathematician known for her work on reciprocity laws in algebraic number theory...
(November 6, 1906-May 7, 2007), a Russian student of his father's, who had begun with work toward an engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
degree but had subsequently switched focus to mathematics, earning her B.A. in 1928. Later that same year, Lehmer married Emma and, following a tour of Northern California and a trip to Japan to meet Emma's family, they moved by car to Providence, Rhode Island
Providence, Rhode Island
Providence is the capital and most populous city of Rhode Island and was one of the first cities established in the United States. Located in Providence County, it is the third largest city in the New England region...
, after Brown University
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...
offered him an instructorship.
Career
Lehmer received a Master's degreeMaster's degree
A master's is an academic degree granted to individuals who have undergone study demonstrating a mastery or high-order overview of a specific field of study or area of professional practice...
and a Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
, both from Brown University, in 1929 and 1930, respectively; his wife obtained a Master's degree in 1930 as well, coaching mathematics to supplement the family income, while also helping her husband type his Ph.D. thesis, An Extended Theory of Lucas' Functions, which he wrote under Jacob Tamarkin
Jacob Tamarkin
Jacob David Tamarkin was a Russian-American mathematician best known for his work in mathematical analysis.- Biography :Tamarkin was born in Chernigov, Imperial Russia to a wealthy family. His father, David Tamarkin, was a physician and his mother, Sophie Krassilschikov, a member of the landed...
.
Movements during the Depression
Lehmer became a National Research Fellow, allowing him to take positions at the California Institute of TechnologyCalifornia Institute of Technology
The California Institute of Technology is a private research university located in Pasadena, California, United States. Caltech has six academic divisions with strong emphases on science and engineering...
from 1930 to 1931 and at Stanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
from 1931 to 1932. In the latter year, the couple's first child Laura was born.
After being awarded a second National Research Fellowship, the Lehmers moved on to Princeton, New Jersey
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
between 1932 and 1934, where Dick spent a short time at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
.
He worked at Lehigh University
Lehigh University
Lehigh University is a private, co-educational university located in Bethlehem, Pennsylvania, in the Lehigh Valley region of the United States. It was established in 1865 by Asa Packer as a four-year technical school, but has grown to include studies in a wide variety of disciplines...
in Pennsylvania
Pennsylvania
The Commonwealth of Pennsylvania is a U.S. state that is located in the Northeastern and Mid-Atlantic regions of the United States. The state borders Delaware and Maryland to the south, West Virginia to the southwest, Ohio to the west, New York and Ontario, Canada, to the north, and New Jersey to...
from 1934 until 1938. Their son Donald was born in 1934 while Dick and Emma were at Lehigh.
The year 1938-1939 was spent in England
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
on a Guggenheim Fellowship
Guggenheim Fellowship
Guggenheim Fellowships are American grants that have been awarded annually since 1925 by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the arts." Each year, the foundation makes...
visiting both the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
and the University of Manchester
University of Manchester
The University of Manchester is a public research university located in Manchester, United Kingdom. It is a "red brick" university and a member of the Russell Group of research-intensive British universities and the N8 Group...
, meeting G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
, John Edensor Littlewood
John Edensor Littlewood
John Edensor Littlewood was a British mathematician, best known for the results achieved in collaboration with G. H. Hardy.-Life:...
, Harold Davenport
Harold Davenport
Harold Davenport FRS was an English mathematician, known for his extensive work in number theory.-Early life:...
, Kurt Mahler
Kurt Mahler
Kurt Mahler was a mathematician and Fellow of the Royal Society.He was a student at the universities in Frankfurt and Göttingen, graduating with a Ph.D...
, Louis Mordell
Louis Mordell
Louis Joel Mordell was a British mathematician, known for pioneering research in number theory. He was born in Philadelphia, USA, in a Jewish family of Lithuanian extraction...
, and Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
. The Lehmers returned to America by ship with second child Donald just before the beginning of the Battle of the Atlantic.
Lehmer continued at Lehigh University for the 1939-1940 academic year.
Settling down
In 1940, Lehmer accepted a position back at the mathematics department of UC Berkeley. At some point in his career there, he developed the Linear congruential generatorLinear congruential generator
A Linear Congruential Generator represents one of the oldest and best-known pseudorandom number generator algorithms. The theory behind them is easy to understand, and they are easily implemented and fast....
(pseudorandom number generator
Pseudorandom number generator
A pseudorandom number generator , also known as a deterministic random bit generator , is an algorithm for generating a sequence of numbers that approximates the properties of random numbers...
), which is frequently referred to as a Lehmer random number generator. The Lehmers also assisted Harry Vandiver
Harry Vandiver
Harry Schultz Vandiver was an American mathematician, known for work in number theory.He was born in Philadelphia, Pennsylvania to John Lyon and Ida Frances Vandiver...
with his work on Fermat's Last Theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
, computing many Bernoulli numbers required.
Lehmer was chairman of the Department of Mathematics at University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
from 1954 until 1957. He continued working at UC Berkeley until 1972, the year he became professor emeritus.
ENIAC involvement
From 1945-1946, Lehmer served on the Computations Committee at Aberdeen Proving Grounds in MarylandMaryland
Maryland is a U.S. state located in the Mid Atlantic region of the United States, bordering Virginia, West Virginia, and the District of Columbia to its south and west; Pennsylvania to its north; and Delaware to its east...
, a group established as part of the Ballistics Research Laboratory to prepare the ENIAC
ENIAC
ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....
for utilization following its completion at the University of Pennsylvania
University of Pennsylvania
The University of Pennsylvania is a private, Ivy League university located in Philadelphia, Pennsylvania, United States. Penn is the fourth-oldest institution of higher education in the United States,Penn is the fourth-oldest using the founding dates claimed by each institution...
's Moore School of Electrical Engineering
Moore School of Electrical Engineering
The Moore School of Electrical Engineering at the University of Pennsylvania came into existence as a result of an endowment from Alfred Fitler Moore on June 4, 1923. It was granted to Penn's School of Electrical Engineering, located in the Towne Building...
; the other Computations Committee members were Haskell Curry
Haskell Curry
Haskell Brooks Curry was an American mathematician and logician. Curry is best known for his work in combinatory logic; while the initial concept of combinatory logic was based on a single paper by Moses Schönfinkel, much of the development was done by Curry. Curry is also known for Curry's...
, Leland Cunningham
Leland Cunningham
Leland Erskin Cunningham was an American astronomer. In a career spanning 50 years, he became an authority on orbit theory and on precise measurements of the orbits of comets, planets, satellites, and space probes...
, and Franz Alt
Franz Alt (mathematician)
Franz Leopold Alt was an Austrian-born American mathematician who made major contributions to computer science in its early days...
. It was during this short tenure that the Lehmers ran some of the first test programs on the ENIAC—according to their academic interests, these tests involved number theory, especially sieve methods
Sieve theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a...
, but also pseudorandom number generation. When they could arrange child care, the Lehmers spent weekends staying up all night running such problems, the first over the Thanksgiving
Thanksgiving
Thanksgiving Day is a holiday celebrated primarily in the United States and Canada. Thanksgiving is celebrated each year on the second Monday of October in Canada and on the fourth Thursday of November in the United States. In Canada, Thanksgiving falls on the same day as Columbus Day in the...
weekend of 1945. (Such tests were run without cost, since the ENIAC would have been left powered on anyway in the interest of minimizing vacuum tube failures.) The problem run during the 3-day Independence Day
Independence Day (United States)
Independence Day, commonly known as the Fourth of July, is a federal holiday in the United States commemorating the adoption of the Declaration of Independence on July 4, 1776, declaring independence from the Kingdom of Great Britain...
weekend of July 4, 1946, with John Mauchly
John Mauchly
John William Mauchly was an American physicist who, along with J. Presper Eckert, designed ENIAC, the first general purpose electronic digital computer, as well as EDVAC, BINAC and UNIVAC I, the first commercial computer made in the United States.Together they started the first computer company,...
serving as computer operator, ran around the clock without interruption or failure. The following Tuesday, July 9, 1946, Lehmer delivered the talk "Computing Machines for Pure Mathematics" as part of the Moore School Lectures
Moore School Lectures
Theory and Techniques for Design of Electronic Digital Computers was a course in the construction of electronic digital computers held at the University of Pennsylvania's Moore School of Electrical Engineering between July 8, 1946 and August 30, 1946, and was the first time any computer topics had...
, in which he introduced computing as an experimental science, and demonstrated the wit and humor typical of his teaching lectures.
Lehmer would remain active in computing developments for the remainder of his career. Upon his return to Berkeley, he made plans for building the California Digital Computer (CALDIC
CALDIC
CALDIC was an electronic digital computer built with the assistance of the Office of Naval Research at the University of California, Berkeley between 1951 and 1955 to assist and enhance research being conducted at the university with a platform for high-speed computing.CALDIC was designed to be...
) with Paul Morton and Leland Cunningham.
McCarthy era
In 1950, Lehmer was one of 31 University of California faculty fired after refusing to sign a loyalty oathLoyalty oath
A loyalty oath is an oath of loyalty to an organization, institution, or state of which an individual is a member.In this context, a loyalty oath is distinct from pledge or oath of allegiance...
, a policy initiated by the Board of Regents of the State of California in 1950 during the Communist scare personified by Senator Joseph McCarthy
Joseph McCarthy
Joseph Raymond "Joe" McCarthy was an American politician who served as a Republican U.S. Senator from the state of Wisconsin from 1947 until his death in 1957...
. Lehmer took a post as Director of the National Bureau of Standards' Institute for Numerical Analysis (INA), working with the Standards Western Automatic Computer (SWAC
SWAC (computer)
The SWAC was an early electronic digital computer built in 1950 by the U.S. National Bureau of Standards in Los Angeles, California. It was designed by Harry Huskey...
). On October 17, 1952, the State Supreme Court proclaimed the oath unconstitutional, and Lehmer returned to Berkeley shortly thereafter.
Later years
Lehmer continued to be active for many years and would certainly qualify as a dotagy, Paul Erdos's term for someone active in their dotage. When John Selfridge was at Northern Illinois University he twice invited Lehmer and Emma to spend a semester there. One year Selfridge arranged that Erdos and Lehmer taught a course together on Research Problems in the Theory of Numbers. Lehmer taught the first eight weeks and then Erdos taught the remainder. Erdos didn't often teach a course, and he said "You know it wasn't that difficult. The only problem was being there."Lehmer had quite a wit. On the occasion of the first Asilomar number theory conference, which became an annual event, Lehmer, as the organizer, was inspecting the facilities of the Asilomar Conference Grounds
Asilomar Conference Grounds
Asilomar Conference Grounds is a conference center built for the YWCA in 1913 at Asilomar State Beach in Pacific Grove, California. Julia Morgan designed and built 16 of the buildings on the property, of which 11 are still standing. It became part of Asilomar State Beach and Conference Grounds in...
--basically a wooden building on the beach. Someone said they couldn't find a blackboard and Lehmer spotted some curtains in the middle of the wall. Moving the curtains aside revealed a very small blackboard, whereupon Lehmer said "Well, I guess we won't be doing any analytic number theory!"
Lasting impact
In addition to his significant contributions to number theory algorithms for multiprecision integers, such as factoring, Euclid's algorithm, long division, and proof of primality, he also formulated Lehmer's conjectureLehmer's conjecture
Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory. Derrick Henry Lehmer conjectured that the Mahler measure of any integral polynomial...
and participated in the Cunningham project
Cunningham project
The Cunningham project aims to find factors of large numbers of the formb^n \pm 1for b = 2, 3, 5, 6, 7, 10, 11, 12 and large exponents n. The project is named after Allan Joseph Champneys Cunningham, who published the first version of the table together with Herbert J. Woodall in 1925...
.
Combinatorics
His father Derrick Norman LehmerDerrick Norman Lehmer
Derrick Norman Lehmer was an American mathematician and number theorist....
, known mainly as a pioneer in number theory computing, also made major contributions to combinatorial computing, having devised algorithms for efficiently generating all the permutations on n elements He also developed two algorithms, rank(p) and unrank(k). Given a permutation p, if k = rank(p), then p is the kth permutation. Given an integer k, unrank(k) is the kth permutation. D N Lehmer's method uses his so called factorial representation of the integer k. See section 5.1 in Permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
.
D. H. Lehmer continued his fathers interest in combinatorial computing and in fact wrote the article "Machine tools of Computation," which is chapter one in the book "Applied Combinatorial Mathematics," by Edwin Beckenbach, 1964. It describes methods for producing permutations, combinations etc. This was a uniquely valuable resource and has only been rivaled recently by Volume 4 of Donald Knuth's series.