Robert Daniel Carmichael (March 1, 1879 – May 2, 1967) was an American
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
.
Biography
Carmichael was born in
Goodwater, Alabama. He attended
Lineville College, briefly, and he earned his
bachelor's degree
A bachelor's degree (from Medieval Latin ''baccalaureus'') or baccalaureate (from Modern Latin ''baccalaureatus'') is an undergraduate degree awarded by colleges and universities upon completion of a course of study lasting three to six years ...
in 1898, while he was studying towards his
Ph.D. degree at
Princeton University
Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...
. Carmichael completed the requirements for his Ph.D. in
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
in 1911. Carmichael's Ph.D. research in mathematics was done under the guidance of the noted American mathematician
G. David Birkhoff, and it is considered to be the first significant American contribution to the knowledge of
differential equations in mathematics.
Carmichael next taught at
Indiana University
Indiana University (IU) is a state university system, system of Public university, public universities in the U.S. state of Indiana. The system has two core campuses, five regional campuses, and two regional centers under the administration o ...
from 1911 to 1915. Then he moved on to the
University of Illinois
The University of Illinois Urbana-Champaign (UIUC, U of I, Illinois, or University of Illinois) is a public university, public land-grant university, land-grant research university in the Champaign–Urbana metropolitan area, Illinois, United ...
, where he remained from 1915 until his retirement in 1947.
Carmichael is known for his research in what are now called the
Carmichael number
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation:
: b^n\equiv b\pmod
for all integers . The relation may also be expressed in the form:
: b^\equiv 1\pmod
for all integers b ...
s (a subset of
Fermat pseudoprimes, numbers satisfying properties of primes described by
Fermat's Little Theorem
In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as
a^p \equiv a \pmod p.
For example, if and , t ...
although they are not primes),
Carmichael's totient function conjecture,
Carmichael's theorem, and the
Carmichael function
In number theory, a branch of mathematics, the Carmichael function of a positive integer is the smallest positive integer such that
:a^m \equiv 1 \pmod
holds for every integer coprime to . In algebraic terms, is the exponent of a group, expone ...
, all significant in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
and in the study of the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s. He found the smallest Carmichael number, 561, and over 50 years later, it was proven that there are infinitely many of them. Carmichael also described the
Steiner system
250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line.
In combinatorial mathematics, a Steiner system (named after Jakob Steiner ...
S(5,8,24) in his 1931 paper ''Tactical Configurations of Rank 2'' and his 1937 book ''Introduction to the
Theory of Groups of Finite Order'', but the structure is often named after
Ernst Witt
Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time.
Biography
Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the f ...
, who rediscovered it in 1938.
While at
Indiana University
Indiana University (IU) is a state university system, system of Public university, public universities in the U.S. state of Indiana. The system has two core campuses, five regional campuses, and two regional centers under the administration o ...
, Carmichael was involved with the
special theory of relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 paper,
"On the Electrodynamics of Moving Bodies", the theory is presen ...
.
Relatives
Carmichael's younger brother was university administrator
Oliver Carmichael
Oliver Cromwell Carmichael (October 3, 1891 – September 25, 1966) served as the third chancellor of Vanderbilt University from 1937 to 1946. He also served as the President of the University of Alabama from 1953 to 1957.
Early life
Oliver Crom ...
.
[CARMICHAEL, Oliver Cromwell]
in ''Who's Who in America
Marquis Who's Who, also known as A.N. Marquis Company ( or ), is an American publisher of a number of directories containing short biographies. The books usually are entitled ''Who's Who in...'' followed by some subject, such as ''Who's Who in A ...
'' (1926 edition); p. 426
Mathematical publications
* ''The Theory of Relativity,'' 1st edition, New York: John Wiley & Sons, Inc., pp. 74, 1913.
* ''The Theory of Numbers,'' New York: John Wiley & Sons, Inc., pp. 94, 1914.
1st edition, New York: John Wiley & Sons, Inc., pp. 118, 1915.
[
]
''The Theory of Relativity.'' 2nd edition
New York: John Wiley & Sons, Inc., pp. 112, 1920.
* ''A Debate on the Theory of Relativity,'' with an introduction by William Lowe Bryan, Chicago: Open Court Pub. CO., pp. 154, 1927.
* ''The calculus,'' Robert D. Carmichael and James H. Weaver, Boston/New York: Ginn & company, pp. 345, 1927.
* ''The Logic of Discovery,'' Chicago/London: Open Court Publishing CO., pp. 280, 1930; Reprinted of Arno press, New York, 1975
* ''Mathematical Tables and Formulas,'' Robert D. Carmichael and Edwin R. Smith, Boston: Ginn & company, pp. 269, 1931; Reprint of Dover Publications, Inc., New York, 1962.
* ''The calculus,'' revised edition by Robert D. Carmichael, James H. Weaver and Lincoln La Paz, Boston/New York: Ginn & company, pp. 384, 1937.
* ''Introduction to the Theory of Groups of finite order,'' Boston/New York: Ginn & company, pp. 447, 1937;
See also
*Pseudoprimes
A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy.
Some sources use the term pseudoprime to ...
Notes and references
External links
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*
MAA presidents: Robert Daniel Carmichael
{{DEFAULTSORT:Carmichael, Robert Daniel
20th-century American mathematicians
American number theorists
University of Illinois faculty
Indiana University Bloomington faculty
Princeton University alumni
Presidents of the Mathematical Association of America
People from Goodwater, Alabama
1879 births
1967 deaths
The American Mathematical Monthly editors