query: Prove $a^n - 1$ is composite
positive_passages.docid: 2522
positive_passages.text: Why is $a^n - b^n$ divisible by $a-b$?
negative_passages.docid: 68824
negative_passages.text: Show that $\frac{(3^{77}-1)}{2}$ is odd and composite
negative_passages.docid: 33344
negative_passages.text: A different way to show that $a^k-1 \mid a^n-1$ if $k\mid n$?
negative_passages.docid: 21781
negative_passages.text: Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$
negative_passages.docid: 119436
negative_passages.text: Proving $2^n\leq 2^{n+1}-2^{n-1}-1$ for all $n\geq 1$ by induction
negative_passages.docid: 97469
negative_passages.text: How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?
negative_passages.docid: 96027
negative_passages.text: Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite.
negative_passages.docid: 152299
negative_passages.text: Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
negative_passages.docid: 82775
negative_passages.text: if $a>1$, Prove that $\lim a^{1\over n}=1$
negative_passages.docid: 4315
negative_passages.text: show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
negative_passages.docid: 133017
negative_passages.text: Prove that $n^n>1\times3\times5\times7\times \dots\times(2n-1)$
negative_passages.docid: 125618
negative_passages.text: Prove $2^{3^n}+1 $ is a multiple of $3^{n+1}$ by induction
negative_passages.docid: 118471
negative_passages.text: Prove that $1-\sum_{i=1}^n a_i < \prod_{i=1}^n(1-a_i) $
negative_passages.docid: 91285
negative_passages.text: Prove $a+b+c+d $ is composite
negative_passages.docid: 43913
negative_passages.text: Prove that if $A^*+A=AA^*$ then $A$ is normal
negative_passages.docid: 89175
negative_passages.text: Proof of $(a+b)^{n+1}$
negative_passages.docid: 75373
negative_passages.text: Show that $2^n>n$ for $n\in \mathbb{N}$