Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of
computational number theory
In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of
computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithm ...
. Lehmer refined
Édouard Lucas
__NOTOC__
François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) 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
Lucas ...
' work in the 1930s and devised the
Lucas–Lehmer test for
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
s. His peripatetic career as a
number theorist
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, with him and his wife taking numerous types of work in the United States and abroad to support themselves during the
Great Depression
The Great Depression (19291939) was an economic shock that impacted most countries across the world. It was a period of economic depression that became evident after a major fall in stock prices in the United States. The economic contagio ...
, fortuitously brought him into the center of research into early electronic computing.
Early life
Lehmer was born in
Berkeley, California
Berkeley ( ) is a city on the eastern shore of San Francisco Bay in northern Alameda County, California, United States. It is named after the 18th-century Irish bishop and philosopher George Berkeley. It borders the cities of Oakland and Emer ...
, to
Derrick Norman Lehmer
Derrick Norman Lehmer (27 July 1867 – 8 September 1938) was an American mathematician and number theorist.
Education
He was educated at the University of Nebraska, obtaining a bachelor's degree in 1893 and master's in 1896. Lehmer was awarded ...
, a professor of mathematics at the
University of California, Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...
, and
Clara Eunice Mitchell.
He studied
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and earned a
Bachelor degree
A bachelor's degree (from Middle Latin ''baccalaureus'') or baccalaureate (from Modern Latin ''baccalaureatus'') is an undergraduate academic degree awarded by colleges and universities upon completion of a course of study lasting three to six ye ...
from UC Berkeley, and continued with graduate studies at the
University of Chicago
The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park neighborhood. The University of Chicago is consistently ranked among the b ...
.
He and his father worked together on
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. The father was a professor of mathematics at the University of California, Berkeley a ...
s.
Marriage
During his studies at Berkeley, Lehmer met
Emma Markovna Trotskaia, a Russian student of his father's, who had begun with work toward an
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
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 is the capital and most populous city of the U.S. state of Rhode Island. One of the oldest cities in New England, it was founded in 1636 by Roger Williams, a Reformed Baptist theologian and religious exile from the Massachusetts Bay ...
, after
Brown University
Brown University is a private research university in Providence, Rhode Island. Brown is the seventh-oldest institution of higher education in the United States, founded in 1764 as the College in the English Colony of Rhode Island and Providenc ...
offered him an
instructorship.
Career
Lehmer received a
Master's degree
A master's degree (from Latin ) is an academic degree awarded by universities or colleges upon completion of a course of study demonstrating mastery or a high-order overview of a specific field of study or area of professional practice. and a
Ph.D.
A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is a ...
, 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 David Tamarkin (russian: Я́ков Дави́дович Тама́ркин, ''Yakov Davidovich Tamarkin''; 11 July 1888 – 18 November 1945) was a Russian-American mathematician best known for his work in mathematical analysis.
Biography ...
.
Movements during the Depression
Lehmer became a
National Research Fellow, allowing him to take positions at the
California Institute of Technology
The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...
from 1930 to 1931 and at
Stanford University
Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is consider ...
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 is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of whi ...
between 1932 and 1934, where Dick spent a short time at the
Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...
.
He worked at
Lehigh University
Lehigh University (LU) is a private research university in Bethlehem, Pennsylvania in the Lehigh Valley region of eastern Pennsylvania. The university was established in 1865 by businessman Asa Packer and was originally affiliated with the Epis ...
in
Pennsylvania
Pennsylvania (; ( Pennsylvania Dutch: )), officially the Commonwealth of Pennsylvania, is a state spanning the Mid-Atlantic, Northeastern, Appalachian, and Great Lakes regions of the United States. It borders Delaware to its southeast, ...
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 is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe b ...
on a
Guggenheim Fellowship
Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...
visiting both the
University of Cambridge
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...
and the
University of Manchester
, mottoeng = Knowledge, Wisdom, Humanity
, established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Univer ...
, meeting
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
,
John Edensor Littlewood
John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...
,
Harold Davenport
Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory.
Early life
Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar Schoo ...
,
Kurt Mahler
Kurt Mahler FRS (26 July 1903, Krefeld, Germany – 25 February 1988, Canberra, Australia) was a German mathematician who worked in the fields of transcendental number theory, diophantine approximation, ''p''-adic analysis, and the geometry of ...
,
Louis Mordell
Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction.
Educatio ...
, and
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
. The Lehmers returned to America by ship with second child Donald just before the beginning of the
Battle of the Atlantic
The Battle of the Atlantic, the longest continuous military campaign in World War II, ran from 1939 to the defeat of Nazi Germany in 1945, covering a major part of the naval history of World War II. At its core was the Allied naval blockade ...
.
Lehmer continued at Lehigh University for the 1939–1940 academic year.
Berkeley
In 1940, Lehmer accepted a position back at the mathematics department of UC Berkeley. Lehmer was chairman of the Department of Mathematics at
University of California, Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...
from 1954 until 1957. He continued working at UC Berkeley until 1972, the year he became
professor emeritus
''Emeritus'' (; female: ''emerita'') is an adjective used to designate a retired chair, professor, pastor, bishop, pope, director, president, prime minister, rabbi, emperor, or other person who has been "permitted to retain as an honorary title ...
.
ENIAC and computer science
From 1945–1946, Lehmer served on the Computations Committee at
Aberdeen Proving Grounds
Aberdeen Proving Ground (APG) (sometimes erroneously called Aberdeen Proving ''Grounds'') is a U.S. Army facility located adjacent to Aberdeen, Harford County, Maryland, United States. More than 7,500 civilians and 5,000 military personnel work at ...
in
Maryland
Maryland ( ) is a state in the Mid-Atlantic region of the United States. It shares borders with Virginia, West Virginia, and the District of Columbia to its south and west; Pennsylvania to its north; and Delaware and the Atlantic Ocean to ...
, a group established as part of the
Ballistics Research Laboratory
The Ballistic Research Laboratory (BRL) was a leading U.S. Army research establishment situated at Aberdeen Proving Ground, Maryland that specialized in ballistics (interior, exterior, and terminal) as well as vulnerability and lethality analysis. ...
to prepare the
ENIAC
ENIAC (; Electronic Numerical Integrator and Computer) was the first programmable, electronic, general-purpose digital computer, completed in 1945. There were other computers that had these features, but the ENIAC had all of them in one packa ...
for utilization following its completion at the
University of Pennsylvania
The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universitie ...
's
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 ...
; the other Computations Committee members were
Haskell Curry
Haskell Brooks Curry (; September 12, 1900 – September 1, 1982) 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 ...
,
Leland Cunningham
Leland Erskin Cunningham (February 10, 1904, in Wiscasset, Maine – May 31, 1989, in Richmond, California) was an American astronomer and discoverer of minor planets. In a career spanning 50 years, he became an authority on orbit theory and on ...
, 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
Thanksgiving is a national holiday celebrated on various dates in the United States, Canada, Grenada, Saint Lucia, Liberia, and unofficially in countries like Brazil and Philippines. It is also observed in the Netherlander town of Leiden and ...
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
An independence day is an annual event commemorating the anniversary of a nation's independence or statehood, usually after ceasing to be a group or part of another nation or state, or more rarely after the end of a military occupation. Man ...
weekend of July 4, 1946, with
John Mauchly
John William Mauchly (August 30, 1907 – January 8, 1980) 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 co ...
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 ''Theory and Techniques for Design of Electronic Digital Computers'' (popularly called the "Moore School Lectures") was a course in the construction of electronic digital computers held at the University of Pennsylvania's Moore School of Electrical ...
, 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 (the California Digital Computer) 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 ...
) with Paul Morton and Leland Cunningham. In September 1949, he presented the
pseudorandom number generator
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generate ...
now known as the
Lehmer random number generator The Lehmer random number generator (named after D. H. Lehmer), sometimes also referred to as the Park–Miller random number generator (after Stephen K. Park and Keith W. Miller), is a type of linear congruential generator (LCG) that oper ...
.
D. H. Lehmer wrote the article "The Machine Tools of Combinatorics," which is the first chapter in
Edwin Beckenbach's ''Applied Combinatorial Mathematics'' (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
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
's series.
The Lehmers also assisted
Harry Vandiver
Harry Schultz Vandiver (21 October 1882 – 9 January 1973) was an American mathematician, known for work in number theory.
He was born in Philadelphia, Pennsylvania to John Lyon and Ida Frances (Everett) Vandiver. He did not complete a form ...
with his work on
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
, using the
Standards Western Automatic Computer
The SWAC (Standards Western Automatic Computer) was an early electronic digital computer built in 1950 by the U.S. National Bureau of Standards (NBS) in Los Angeles, California. It was designed by Harry Huskey.
Overview
Like the SEAC which w ...
to do many calculations involving
Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
.
McCarthy era
In 1950, Lehmer was one of 31 University of California faculty fired after refusing to sign a
loyalty oath
A loyalty oath is a pledge of allegiance to an organization, institution, or state of which an individual is a member. In the United States, such an oath has often indicated that the affiant has not been a member of a particular organization or ...
, 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 Raymond McCarthy (November 14, 1908 – May 2, 1957) was an American politician who served as a Republican U.S. Senator from the state of Wisconsin from 1947 until his death in 1957. Beginning in 1950, McCarthy became the most visi ...
. Lehmer took a post as Director of the
National Bureau of Standards
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
'
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.
Later years
Lehmer continued to be active for many years. When
John Selfridge
John Lewis Selfridge (February 17, 1927 – October 31, 2010), was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics.
Education
Selfridge received his Ph.D. in 195 ...
was at Northern Illinois University he twice invited Lehmer and Emma to spend a semester there. One year Selfridge arranged that Erdős and Lehmer taught a course together on Research Problems in the Theory of Numbers. Lehmer taught the first eight weeks and then Erdős taught the remainder. Erdős 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 (now called
West Coast Number Theory
West Coast Number Theory (WCNT), a meeting that has also been known variously as the Western Number Theory Conference and the Asilomar Number Theory meeting, is an annual gathering of number theorists first organized by D. H. and Emma Lehmer at t ...
), Lehmer, as the organizer, was inspecting the facilities of the
Asilomar Conference Grounds
Asilomar Conference Grounds is a conference center built for the Young Women's Christian Association (YWCA). It is located east of what was known as Moss Beach on the western tip of the Monterey Peninsula in Pacific Grove, California. Between 1913 ...
—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 conjecture
Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant \mu>1 such that every polynomial with integer coeffic ...
and participated in the
Cunningham project.
Death
Lehmer died in Berkeley on May 22, 1991.
See also
*
Lucas–Lehmer–Riesel test
In mathematics, the Lucas–Lehmer–Riesel test is a primality test for numbers of the form ''N'' = ''k'' ⋅ 2''n'' − 1 ( Riesel numbers) with odd ''k'' < 2''n''. The test was developed by Hans ...
*
Størmer's theorem
In number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equ ...
References
External links
*
Photo of Derrick Henry Lehmer
*
* [https://web.archive.org/web/20060714111222/http://infoshare1.princeton.edu/libraries/firestone/rbsc/finding_aids/mathoral/pmc12.htm Interview with the Lehmers and others about their experiences at Princeton]
*
*
{{DEFAULTSORT:Lehmer, D. H.
1905 births
1991 deaths
20th-century American mathematicians
Number theorists
Institute for Advanced Study visiting scholars
University of California, Berkeley alumni
University of California, Berkeley faculty
University of Chicago alumni
Brown University alumni
Stanford University Department of Mathematics faculty
Lehigh University faculty