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.

Printable View

- Jan 14th 2007, 09:05 PMJenny20carmichael numbers
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. - Jan 15th 2007, 07:01 AMThePerfectHacker
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.

----

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.

---

I am supprised, where did you get this theorem from? First time I hear of it. It is really interesting. - May 9th 2009, 06:52 PMMedia_ManRewriteQuote:

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