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A composite natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101001.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101002.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101003.png" />, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101004.png" /> is relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101005.png" />. Thus they are pseudo-primes (cf. [[Pseudo-prime|Pseudo-prime]]) for every such base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101006.png" />. These numbers play a role in the theory of probabilistic primality tests (cf. [[Probabilistic primality test|Probabilistic primality test]]), as they show that Fermat's theorem, to wit $ a^p \equiv a $ modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101008.png" />, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101009.png" /> is prime and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010010.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010011.png" />, is not a sufficient criterion for primality (cf. also [[Fermat little theorem|Fermat little theorem]]).
 
A composite natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101001.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101002.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101003.png" />, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101004.png" /> is relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101005.png" />. Thus they are pseudo-primes (cf. [[Pseudo-prime|Pseudo-prime]]) for every such base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101006.png" />. These numbers play a role in the theory of probabilistic primality tests (cf. [[Probabilistic primality test|Probabilistic primality test]]), as they show that Fermat's theorem, to wit $ a^p \equiv a $ modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101008.png" />, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c1101009.png" /> is prime and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010010.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010011.png" />, is not a sufficient criterion for primality (cf. also [[Fermat little theorem|Fermat little theorem]]).
  
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010022.png" /> denote the number of Carmichael numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010023.png" />. W.R. Alford, A. Granville and C. Pomerance [[#References|[a1]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010024.png" /> for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010025.png" />. This settled a long-standing conjecture that there are infinitely many Carmichael numbers. It is believed on probabilistic grounds that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010026.png" /> [[#References|[a4]]].
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010022.png" /> denote the number of Carmichael numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010023.png" />. W.R. Alford, A. Granville and C. Pomerance [[#References|[a1]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010024.png" /> for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010025.png" />. This settled a long-standing conjecture that there are infinitely many Carmichael numbers. It is believed on probabilistic grounds that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010026.png" /> [[#References|[a4]]].
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P. Erdős proved in 1956 that $ C(X) < X.\exp(- k \log X \log\log\log X /  \log\log X) $ for some constant $ k $: he also gave a heuristic suggesting that  his upper bound should be close to the true rate of growth of $ C(X) $.[[#References|[a5]]]
  
 
There is apparently no better way to compute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010027.png" /> than to make a list of the Carmichael numbers up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010028.png" />. The most exhaustive computation to date (1996) is that of R.G.E. Pinch, who used the methods of [[#References|[a3]]] to determine that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010029.png" />.
 
There is apparently no better way to compute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010027.png" /> than to make a list of the Carmichael numbers up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010028.png" />. The most exhaustive computation to date (1996) is that of R.G.E. Pinch, who used the methods of [[#References|[a3]]] to determine that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010029.png" />.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.R. Alford,  A. Granville,  C. Pomerance,  "There are infinitely many Carmichael numbers"  ''Ann. of Math.'' , '''140'''  (1994)  pp. 703–722</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.D. Carmichael,  "Note on a new number theory function"  ''Bull. Amer. Math. Soc.'' , '''16'''  (1910)  pp. 232–238  (See also: Amer. Math. Monthly 19 (1912), 22–27)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.G.E. Pinch,  "The Carmichael numbers up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010030.png" />"  ''Math. Comp.'' , '''61'''  (1993)  pp. 381–391</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Pomerance,  J.L. Selfridge,  S.S. Wagstaff, Jr.,  "The pseudoprimes to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010031.png" />"  ''Math. Comp.'' , '''35'''  (1980)  pp. 1003–1026</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W.R. Alford,  A. Granville,  C. Pomerance,  "There are infinitely many Carmichael numbers"  ''Ann. of Math.'' , '''140'''  (1994)  pp. 703–722</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.D. Carmichael,  "Note on a new number theory function"  ''Bull. Amer. Math. Soc.'' , '''16'''  (1910)  pp. 232–238  (See also: Amer. Math. Monthly 19 (1912), 22–27)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  R.G.E. Pinch,  "The Carmichael numbers up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010030.png" />"  ''Math. Comp.'' , '''61'''  (1993)  pp. 381–391</TD></TR>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Pomerance,  J.L. Selfridge,  S.S. Wagstaff, Jr.,  "The pseudoprimes to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110100/c11010031.png" />"  ''Math. Comp.'' , '''35'''  (1980)  pp. 1003–1026</TD></TR>
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<TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Erdős, "On pseudoprimes and Carmichael numbers" ''Publ. Math. Debrecen', '''4''' (1956) pp.201–206.</TD></TR>
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</table>

Revision as of 21:43, 21 February 2013


A composite natural number for which modulo , whenever is relatively prime to . Thus they are pseudo-primes (cf. Pseudo-prime) for every such base . These numbers play a role in the theory of probabilistic primality tests (cf. Probabilistic primality test), as they show that Fermat's theorem, to wit $ a^p \equiv a $ modulo , whenever is prime and modulo , is not a sufficient criterion for primality (cf. also Fermat little theorem).

The first five Carmichael numbers are

R.D. Carmichael [a2] characterized them as follows. Let be the exponent of the multiplicative group of integers modulo , that is, the least making all th powers in the group equal to . (This is readily computed from the prime factorization of .) Then a composite natural number is Carmichael if and only if . From this it follows that every Carmichael number is odd, square-free, and has at least distinct prime factors.

Let denote the number of Carmichael numbers . W.R. Alford, A. Granville and C. Pomerance [a1] proved that for sufficiently large . This settled a long-standing conjecture that there are infinitely many Carmichael numbers. It is believed on probabilistic grounds that [a4].

P. Erdős proved in 1956 that $ C(X) < X.\exp(- k \log X \log\log\log X / \log\log X) $ for some constant $ k $: he also gave a heuristic suggesting that his upper bound should be close to the true rate of growth of $ C(X) $.[a5]

There is apparently no better way to compute than to make a list of the Carmichael numbers up to . The most exhaustive computation to date (1996) is that of R.G.E. Pinch, who used the methods of [a3] to determine that .

References

[a1] W.R. Alford, A. Granville, C. Pomerance, "There are infinitely many Carmichael numbers" Ann. of Math. , 140 (1994) pp. 703–722
[a2] R.D. Carmichael, "Note on a new number theory function" Bull. Amer. Math. Soc. , 16 (1910) pp. 232–238 (See also: Amer. Math. Monthly 19 (1912), 22–27)
[a3] R.G.E. Pinch, "The Carmichael numbers up to " Math. Comp. , 61 (1993) pp. 381–391
[a4] C. Pomerance, J.L. Selfridge, S.S. Wagstaff, Jr., "The pseudoprimes to " Math. Comp. , 35 (1980) pp. 1003–1026
[a5] P. Erdős, "On pseudoprimes and Carmichael numbers" Publ. Math. Debrecen', 4 (1956) pp.201–206.
How to Cite This Entry:
Carmichael number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carmichael_number&oldid=29456
This article was adapted from an original article by E. Bach (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article