Ive been trying to prove that the number 6601 is a Carmichael number. Ive gone some way to prove it but I don't like it. The first thing I did was look up the prime factors of 6601. And they are(adsbygoogle = window.adsbygoogle || []).push({});

[tex]6601 = 7 \times 23 \times 41[/tex]

And then I noticed that for each prime factor [itex]p_i = \{7,23,41\}[/itex] we have

[tex]p_i - 1 = n \quad \mbox{and } n | (6601 - 1)[/tex]

So that 7 - 1 = 6 and 6 divides 6600, 22 divides 6600, and 40 divides 6600.

Now, Fermat's Little Theorem says that if a is an integer and q is coprime to a, then q divides [itex]a^{q-1} - 1[/itex]. And from this we can say that

[tex]a^{q-1} \equiv 1(\mod p_i)[/tex]

So since 7-1 divides 6601-1 we can say that

[tex]a^{6600} \equiv 1(\mod 7)[/tex]

[tex]a^{6600} \equiv 1(\mod 23)[/tex]

[tex]a^{6600} \equiv 1(\mod 41)[/tex]

because 7, 23, and 41 all divide q-1. and a and q are coprime. Multiplying these together we get

[tex]a^{6600} \equiv 1(\mod 6601) \quad \forall a\in\mathbb{Z}[/tex]

But is this enough to prove that 6601 is a Carmichael Number?

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# Carmichael Numbers

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