# Questions tagged [carmichael-numbers]

Use this tag for questions about composite numbers n that satisfy b^(n-1) ≡ 1 (mod n) for all integers b relatively prime to n.

12 questions
41 views

### Wrong wolframalpha result, calculating Carmichael number?

I put to wolframalpha $3^{560}\pmod{561}$ and result should be $1$ ($561$ is Carmichael number), but result is $375.$ https://www.wolframalpha.com/input/?i=3%5E560mod561 Why this happens? I found ...
32 views

### Lower bounds for totient function of a Carmichael number

Short version: I am wondering if there are any good bounds of the form $\phi(n) \geq f(n)\cdot n$ with $f(n)$ close to 1 for high $n$, optionally under the assumption that $n$ is a Carmichael number. ...
33 views

### Miller’s test for the base $b$

Definition: Let $n$ be an integer with $n > 2$ and $n − 1 = 2^st$, where $s$ is a non-negative integer and $t$ is an odd positive integer. We say that $n$ passes Miller’s test for the base $b$ if ...
53 views

### Large Carmichael number

I need to find or generate a very large Carmichael number (500 digits or longer). I've tried to find a database or just an example of such a number but failed. Is there any examples of really big ...
76 views

### Fermat's Little Theorem and Carmichael Numbers

Fermat's little theorem states that if $p$ is a prime number and $a$ is a positive integer, then $p|a^p-a$. However, the converse is false, that is, for integers $a$ and $p$, if $p|a^p-a$, then $a$ ...
31 views

### Carmicheal numbers are square free

Carmichael number square free I was reading this question.can some one explain how to arrive from here In particular, $a^{n−1} \equiv 1\pmod{p^t}$, to here $a^n \equiv a\pmod{p^2}$. Thank you
42 views

### Can a finite set of bases guarantee that a number is Carmichael or prime?

If a positive integer $N>1$ is a Carmichael number, it passes the weak Fermat-pseudoprime-test for every base coprime to it. I wonder whether the converse is true in the following sense : Is ...
53 views

### Reducing modulo $15841$

I want reduce the following congruence $17^{15842}\pmod{15841}$. My first stab at this was to use Fermat's Little Theorem, but then I realized that $15841$ is not a prime number. In fact, 15841 ...
### Every integer in the form $(6m + 1)(12m + 1)(18m + 1)$, is a carmichael number.
Question: Show that every integer of the form $(6m + 1)(12m + 1)(18m + 1)$, where m is a positive integer such that $6m + 1$, $12m + 1$, and $18m + 1$ are all primes, is a Carmichael Number I know ...