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
Goodwater is a town in Coosa County, Alabama, United States. At the 2020 census, the population was 1,291. It is part of the Talladega-Sylacauga Micropolitan Statistical Area.
Geography
Goodwater is located near the northeast corner of Coosa C ...
. He attended
Lineville College, briefly, and he earned his
bachelor's 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 ...
in 1898, while he was studying towards his
Ph.D. degree at
Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...
. Carmichael completed the requirements for his Ph.D. in
mathematics 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 equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
s in mathematics.
Carmichael next taught at
Indiana University
Indiana University (IU) is a system of public universities in the U.S. state of Indiana.
Campuses
Indiana University has two core campuses, five regional campuses, and two regional centers under the administration of IUPUI.
* Indiana Univers ...
from 1911 to 1915. Then he moved on to the
University of Illinois
The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the Unive ...
, 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 n, which in modular arithmetic satisfies the congruence relation:
:b^n\equiv b\pmod
for all integers b. The relation may also be expressed in the form:
:b^\equiv 1\pmod.
for all integers ...
s (a subset of
Fermat pseudoprime
In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.
Definition
Fermat's little theorem states that if ''p'' is prime and ''a'' is coprime to ''p'', then ''a'p''� ...
s, numbers satisfying properties of primes described by
Fermat's Little Theorem although they are not primes),
Carmichael's totient function conjecture,
Carmichael's theorem, and the
Carmichael function, all significant in
number theory
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 ...
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 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 because the only way ...
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 Steine ...
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 ...
, who rediscovered it in 1938.
While at
Indiana University
Indiana University (IU) is a system of public universities in the U.S. state of Indiana.
Campuses
Indiana University has two core campuses, five regional campuses, and two regional centers under the administration of IUPUI.
* Indiana Univers ...
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 regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
.
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 ...
Notes and references
External links
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*
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*
MAA presidents: Robert Daniel Carmichael
{{DEFAULTSORT:Carmichael, Robert Daniel
20th-century American mathematicians
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