question
Suppose that p, 2p-1, 3p-2 are all primes, with p>3 . Prove that p(2p-1)(3p-2) is a Carmichael number. Find the smallest Carmichael number of this form.
Thank you very much.
First I need to explain what a Carmichael number is.
Any prime satisfiers this (Fermat's little theorem):
For any integer .
However, the converse fails.
For example,
.
This is an example of a pseudoprime (to base two) for which the converse fails.
A pseudoprime to all base is call a Carmichaeal number.
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Since are all primes we have.
Let .
Then, Fermat's Elegant theorem (different version than above)
Thus,
Important Note is an integer because primes are odd (usually) and we require for to be an integer we need that .
Now, because of relative primeness in the moduli we have,
Thus,
.
Thus, if .
And, are primes.
Then,
is a Carmichael number.
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I am supprised, where did you get this theorem from? First time I hear of it. It is really interesting.
Substitute , so . QED.Suppose that p, 2p-1, 3p-2 are all primes, with p>3 . Prove that p(2p-1)(3p-2) is a Carmichael number.
Carmichael Number -- from Wolfram MathWorld