Carmichael numbers in other bases

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Carmichael numbers exhibit the property that for any base \( a \) coprime to the number, \( a^{n-1} \equiv 1 \mod n \) holds true, according to Fermat's Little Theorem. The discussion clarifies that for the Carmichael number 561, this property applies to all bases except its prime factors, which are 3, 11, and 17. The confusion arises from the distinction between the base of the exponent and the base of notation. Ultimately, Carmichael numbers should return 1 for bases that are coprime to them, reinforcing their unique mathematical characteristics. Understanding this distinction is crucial for correctly applying Fermat's theorem to Carmichael numbers.
?uestionable
Does anyone know if a Carmichael number returns 1 if you use Fermat's Little Theorem on it base 3?
 
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Yeah, it's the same. Why would it be different?
 
Carmichael numbers in different bases

Iwas at this site:

http://www.newnation.ca/sniffy/fermat.htm

and it had the example 561= 3x11x17
and stated that except for the bases 3, 11 and 17 A^560 = 1 (mod 561), its this statement that has me confused. I was under the impression that Carmichael numbers returned 1 in all bases.
 
Oh, base of the exponent, not the base of notation.
Should still be 1 though.
 
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