**Robert Carmichael** was an American number theorist best known for discovering what are now called Carmichael numbers.

- Carmichael attended Lineville College in Lineville, Clay County, Alabama and graduated with an A.B. in 1898.
- Carmichael trained as a Presbyterian Minister and was living in Hartselle, Alabama, in early 1905 when he began submitting problems to the American Mathematical Monthly.
- Following the award of his doctorate, Carmichael was appointed as an Assistant Professor of Mathematics at Indiana University.
- Carmichael was promoted to Associate Professor at Indiana in 1912 and in the autumn of that year he delivered a short course on the theory of relativity.
- Based on these papers and his lecture course, Carmichael published a 74-page book The theory of relativity in 1913.
- Of course this work was on what is now called the special theory of relativity and after Einstein published the general theory of relativity, Carmichael brought out a second edition of his book in 1920.
- Between the publication of the two editions, Carmichael published two further books: The Theory of Numbers (1914) and Diophantine analysis (1915).
- The 'proof' Carmichael had of this result appeared in the paper On Euler's φ-function that he published in 1907.
- Whether the conjecture is true is still an open question, now known as Carmichael's conjecture, despite considerable efforts being made by many mathematicians.
- In 1915 Carmichael left Indiana University.
- We have already mentioned several of the outstanding books written by Carmichael.
- In May 1926 a debate on the theory of relativity was held at Indiana University and Carmichael both participated in the debate and edited the resulting volume A Debate on the Theory of Relativity (1927).
- Also in 1927, in collaboration with James Henry Weaver (1883-1942), Carmichael published The Calculus.
- In 1930 Carmichael, in collaboration with Edwin R Smith, published Plane and Spherical Trigonometry based on lectures the authors had given.
- Also in 1930, Carmichael published The Logic of Discovery.
- One thus derives unusual satisfaction in the perusal of Professor Carmichael's book since one is never on a single page uncertain as to the meaning of the author.
- In 1931 Mathematical Tables and Formulas, compiled by Carmichael and Edwin Smith, appeared.
- Finally, let us mention another outstanding text by Carmichael Introduction to the Theory of Groups of Finite Order.
- The reader of this biography may have heard of Carmichael only because of the Carmichael numbers, and have wondered why they have not yet been mentioned.
- A 'Carmichael number' is a non-prime nnn satisfying this condition for any xxx coprime to nnn.
- It was given this name since Carmichael discovered the first such number 561 in 1910.
- For many years it was an open problem as to whether there were infinitely many Carmichael numbers, but this was settled in 1994 by W R Alford, A Granville, and C Pomerance in their paper There are infinitely many Carmichael numbers.
- Carmichael was elected a fellow of the American Association for the Advancement of Science.
- Carmichael was Vice President of Section A in 1934.
- In addition to the very considerable mathematical output by Carmichael, he also served the mathematical community by undertaking editorial duties.

Born 1 March 1879, Goodwater, Coosa County, Alabama, USA. Died 2 May 1967, Merriam, Northeast Johnson County, Kansas, USA.

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Astronomy, Origin Usa

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive