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, structure, space, models, and change. History On ...

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 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 ...

Princeton University Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

. Carmichael completed the requirements for his Ph.D. in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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, an ...

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 Universit ...

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 University ...

, 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 Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = ...

although they are not primes),

Carmichael's totient function conjecture In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function ''φ''(''n''), which counts the number of integers less than and coprime to ''n''. It states that, for every ''n'' there ...

Carmichael's theorem In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind ''U'n''(''P'', ''Q'') with relatively prime parameters ''P'', ...

, 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 the multip ...

, 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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

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 ways ...

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 ...

, who rediscovered it in 1938. While at

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 Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...

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.
''Diophantine analysis''
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 Lincoln LaPaz (February 12, 1897 – October 19, 1985) was an American astronomer from the University of New Mexico and a pioneer in the study of meteors. Early life and education He was born in Wichita, Kansas on February 12, 1897 to Charles Me ...

, 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.

Notes and references

External links

* * * *
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

Biography

Mathematical publications

See also

Notes and references

External links