Robert Daniel Carmichael
Encyclopedia
Robert Daniel Carmichael (March 1, 1879 – May 2, 1967) was a leading American mathematician
. Carmichael was born in Goodwater, Alabama
. He attended Lineville College, briefly, and he earned his bachelor's degree
in 1898, while he was studying towards his Ph.D. degree at Princeton University
. 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
s in mathematics.
Carmichael next taught at Indiana University
from 1911 to 1915. Then he moved on to the University of Illinois
, where he remained from 1915 until his retirement in 1947.
Carmichael is known for his mathematical research in what are now called the Carmichael numbers (numbers satisfying properties of primes described by Fermat's Little Theorem
although they are not primes), Carmichael's theorem
, and the Carmichael function, all significant in number theory
and in the study of the prime number
s. Carmichael might have been the first to describe the Steiner system
S(5,8,24), a structure often attributed to Ernst Witt
. Carmichael's research into Fermat's Little Theorem established the study of the set of Fermat pseudoprimes. Carmichael found the smallest one, and over 50 years later, it was proven that there are infinitely-many of them.
While at Indiana University
Carmichael was involved with special theory of relativity.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. Carmichael was born in Goodwater, Alabama
Goodwater, Alabama
Goodwater is a city in Coosa County, Alabama, United States. At the 2010 census the population was 1,475. It is part of the Alexander City Micropolitan Statistical Area.-Geography:Goodwater is located at .According to the U.S...
. He attended Lineville College, briefly, and he earned his bachelor's degree
Bachelor's degree
A bachelor's degree is usually an academic degree awarded for an undergraduate course or major that generally lasts for three or four years, but can range anywhere from two to six years depending on the region of the world...
in 1898, while he was studying towards his Ph.D. degree at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. Carmichael completed the requirements for his Ph.D. in 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...
in 1911. Carmichael's Ph.D. research in mathematics was done under the guidance of the noted American mathematician G. David Birkhoff
George David Birkhoff
-External links:* − from National Academies Press, by Oswald Veblen....
, and it is considered to be the first significant American contribution to the knowledge of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s in mathematics.
Carmichael next taught at Indiana University
Indiana University Bloomington
Indiana University Bloomington is a public research university located in Bloomington, Indiana, in the United States. IU Bloomington is the flagship campus of the Indiana University system. Being the flagship campus, IU Bloomington is often referred to simply as IU or Indiana...
from 1911 to 1915. Then he moved on to the University of Illinois
University of Illinois at Urbana-Champaign
The University of Illinois at Urbana–Champaign is a large public research-intensive university in the state of Illinois, United States. It is the flagship campus of the University of Illinois system...
, where he remained from 1915 until his retirement in 1947.
Carmichael is known for his mathematical research in what are now called the Carmichael numbers (numbers satisfying properties of primes described by Fermat's Little Theorem
Fermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...
although they are not primes), Carmichael's theorem
Carmichael's theorem
Carmichael's theorem, named after the American mathematician R.D. Carmichael, states that for n greater than 12, the nth Fibonacci number F has at least one prime factor that is not a factor of any earlier Fibonacci number. The only exceptions for n up to 12 are:Carmichael's theorem, named after...
, and the Carmichael function, all significant in number theory
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...
and in the study of the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s. Carmichael might have been the first to describe the Steiner system
Steiner system
250px|right|thumbnail|The [[Fano plane]] is an S Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line....
S(5,8,24), a structure often attributed to Ernst Witt
Ernst Witt
Ernst Witt was a German mathematician born on the island of Als . Shortly after his birth, he and his parents moved to China, and he did not return to Europe until he was nine....
. Carmichael's research into Fermat's Little Theorem established the study of the set of Fermat pseudoprimes. Carmichael found the smallest one, and over 50 years later, it was proven that there are infinitely-many of them.
While at Indiana University
Indiana University Bloomington
Indiana University Bloomington is a public research university located in Bloomington, Indiana, in the United States. IU Bloomington is the flagship campus of the Indiana University system. Being the flagship campus, IU Bloomington is often referred to simply as IU or Indiana...
Carmichael was involved with special theory of relativity.
Mathematical publications
- The Theory of Relativity, 1.edition, New York: John Wiley & Sons, Inc., pp. 74, 1913.
- The Theory of Numbers, New York: John Wiley & Sons, Inc., pp. 94, 1914.
- Diophantine analysis, 1.edition, New York: John Wiley & Sons, Inc., pp. 118, 1915.
- The Theory of Relativity. 2.edition, New York: John Wiley & Sons, Inc., pp. 112, 1920.
- A Debate on the Theory of Relativity, with an introduction of 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 expenditure of 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; Reprint of Dover Publications, Inc., New York, 1956.
External links
- Works by Robert D. Carmichael, at Project GutenbergProject GutenbergProject Gutenberg is a volunteer effort to digitize and archive cultural works, to "encourage the creation and distribution of eBooks". Founded in 1971 by Michael S. Hart, it is the oldest digital library. Most of the items in its collection are the full texts of public domain books...