Carmichael number information

In number theory, a Carmichael number is a composite number $n$ which in modular arithmetic satisfies the congruence relation:

b^{n}\equiv b{\pmod {n}}

for all integers $b$ .^[1] The relation may also be expressed^[2]^{[failed verification]} in the form:

b^{n-1}\equiv 1{\pmod {n}}

for all integers $b$ that are relatively prime to $n$ . They are infinite in number.^[3]

They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of primality.^[4]

The Carmichael numbers form the subset K₁ of the Knödel numbers.

The Carmichael numbers were named after the American mathematician Robert Carmichael by Nicolaas Beeger, in 1950. Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "F numbers" for short^[5].

^ Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. Vol. 126 (second ed.). Boston, MA: Birkhäuser. ISBN 978-0-8176-3743-9. Zbl 0821.11001.
^ Crandall, Richard; Pomerance, Carl (2005). Prime Numbers: A Computational Perspective (second ed.). New York: Springer. p. 133. ISBN 978-0387-25282-7.
^ W. R. Alford; Andrew Granville; Carl Pomerance (1994). "There are Infinitely Many Carmichael Numbers" (PDF). Annals of Mathematics. 140 (3): 703–722. doi:10.2307/2118576. JSTOR 2118576. Archived (PDF) from the original on 2005-03-04.
^ Cepelewicz, Jordana (13 October 2022). "Teenager Solves Stubborn Riddle About Prime Number Look-Alikes". Quanta Magazine. Retrieved 13 October 2022.
^ Ore, Øystein (1948). Number Theory and Its History. New York: McGraw-Hill. pp. 331–332 – via Internet Archive.

[1] Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. Vol. 126 (second ed.). Boston, MA: Birkhäuser. ISBN 978-0-8176-3743-9. Zbl 0821.11001.

[2] Crandall, Richard; Pomerance, Carl (2005). Prime Numbers: A Computational Perspective (second ed.). New York: Springer. p. 133. ISBN 978-0387-25282-7.

[Alford-1994-3] W. R. Alford; Andrew Granville; Carl Pomerance (1994). "There are Infinitely Many Carmichael Numbers" (PDF). Annals of Mathematics. 140 (3): 703–722. doi:10.2307/2118576. JSTOR 2118576. Archived (PDF) from the original on 2005-03-04.

[Cepelewicz-2022-4] Cepelewicz, Jordana (13 October 2022). "Teenager Solves Stubborn Riddle About Prime Number Look-Alikes". Quanta Magazine. Retrieved 13 October 2022.

[5] Ore, Øystein (1948). Number Theory and Its History. New York: McGraw-Hill. pp. 331–332 – via Internet Archive.

Carmichael number information

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