query: Suppose k is an integer such that $6k + 1$, $12k + 1$, $18k + 1$ are primes. Show that $n = (6k + 1)(12k + 1)(18k + 1)$ is a Carmichael number.
positive_passages.docid: 152299
positive_passages.text: Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
negative_passages.docid: 65757
negative_passages.text: Prove that $n! + k$ is a composite number
negative_passages.docid: 6700
negative_passages.text: An Olympiad problem in prime numbers :can we proof that "n" is prime in this case "
negative_passages.docid: 71880
negative_passages.text: Prove that every prime factor of $12x^2+1$ has the form $6n+1$
negative_passages.docid: 123033
negative_passages.text: Showing a number $n$ is prime if $(n-2)! \equiv 1 \pmod n$
negative_passages.docid: 23544
negative_passages.text: Complete induction proof that every $n > 1$ can be written as a product of primes
negative_passages.docid: 43939
negative_passages.text: Show that $3^{2n+1}-4^{n+1}+6^n$ is never prime for natural n except 1.
negative_passages.docid: 102064
negative_passages.text: Prime number between $n$ and $n!+1$
negative_passages.docid: 111037
negative_passages.text: Consider numbers of the type $n=2^m+1$. Prove that such an n is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime.
negative_passages.docid: 40968
negative_passages.text: show if $n=4k+3$ is a prime and ${a^2+b^2} \equiv 0 \pmod n$ , then $a \equiv b \equiv 0 \pmod n$
negative_passages.docid: 5142
negative_passages.text: Infinitely many primes of the form $pn+1$
negative_passages.docid: 116970
negative_passages.text: On the number divisors of a number
negative_passages.docid: 125188
negative_passages.text: how to prove $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod{mn}$ where m and n are relatively prime?
negative_passages.docid: 113763
negative_passages.text: Let $p$,$\ q$ be prime numbers and $n\in \mathbb{N}$ such that $p \nmid n-1$. If $p \mid n^q-1$, then $q \mid p-1$.
negative_passages.docid: 2218
negative_passages.text: Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem]
negative_passages.docid: 91069
negative_passages.text: Express an even number as a sum of primes
negative_passages.docid: 6918
negative_passages.text: Primes congruent to 1 mod 6