Difference between revisions of "Carmichael number"

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==Carmichael numbers==
 
==Carmichael numbers==
  
A [[Carmichael number]] is a [[composite number]]s that satisfies [[Fermat's Little Theorem]], <math>a^p \equiv a \pmod{p}.</math>or <math>a^{p - 1} \equiv 1 \pmod{p}.</math> In this case, <math>p</math> is the Carmichael number.
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A [[Carmichael number]] is a [[composite number]] that satisfies [[Fermat's Little Theorem]], <math>a^p \equiv a \pmod{p}.</math>or <math>a^{p - 1} \equiv 1 \pmod{p}.</math> In this case, <math>p</math> is the Carmichael number.
  
The first <math>7</math> are:
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The first <math>7</math> Carmichael numbers are:
  
\begin{align}
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<cmath>\begin{align}
 
561 &= 3 \cdot 11 \cdot 17 \
 
561 &= 3 \cdot 11 \cdot 17 \
 
1105 &= 5 \cdot 13 \cdot 17 \
 
1105 &= 5 \cdot 13 \cdot 17 \
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6601 &= 7 \cdot 23 \cdot 41 \
 
6601 &= 7 \cdot 23 \cdot 41 \
 
8991 &= 7 \cdot 19 \cdot 67.
 
8991 &= 7 \cdot 19 \cdot 67.
\end{align}
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\end{align}</cmath>
  
 
==See Also==
 
==See Also==
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* [[Carmichael function]]
 
* [[Carmichael function]]
  
~ [[User:Enderramsby]]
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~ [[User:Enderramsby|enderramsby]]
  
  
 
{{stub}}
 
{{stub}}
  
[[Category:Number Theory]]
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[[Category:Number theory]]

Latest revision as of 14:52, 3 August 2022

Carmichael numbers

A Carmichael number is a composite number that satisfies Fermat's Little Theorem, $a^p \equiv a \pmod{p}.$or $a^{p - 1} \equiv 1 \pmod{p}.$ In this case, $p$ is the Carmichael number.

The first $7$ Carmichael numbers are:

\begin{align} 561 &= 3 \cdot 11 \cdot 17 \\ 1105 &= 5 \cdot 13 \cdot 17 \\ 1729 &= 7 \cdot 13 \cdot 19 \\ 2465 &= 5 \cdot 17 \cdot 29 \\ 2821 &= 7 \cdot 13 \cdot 31 \\ 6601 &= 7 \cdot 23 \cdot 41 \\ 8991 &= 7 \cdot 19 \cdot 67. \end{align}

See Also


~ enderramsby


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