query: Prove that $89|2^{44}-1$
positive_passages.docid: 28957
positive_passages.text: Verifying that $2^{44}-1$ is divisible by $89$
negative_passages.docid: 157294
negative_passages.text: Prove that if $n>1$ is odd, then $1^n+2^n+3^n+\ldots+(n-1)^n$ is divisible by $n$
negative_passages.docid: 83662
negative_passages.text: Prove that $ a $ divides $1^{a^{n}} + 2^{a^{n}} + ... + (a-1)^{a^{n}}$?
negative_passages.docid: 33996
negative_passages.text: Proving that $\gcd(2^m - 1, 2^n - 1) = 2^{\gcd(m,n )} - 1$
negative_passages.docid: 110826
negative_passages.text: prove that : $ i = \sqrt {-1}\ $
negative_passages.docid: 35167
negative_passages.text: Prove congruence using fermat's thm
negative_passages.docid: 128415
negative_passages.text: Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\dots+\frac{1}{3001}<\frac43$
negative_passages.docid: 26741
negative_passages.text: How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
negative_passages.docid: 24469
negative_passages.text: Prove $ 2<(1+\frac{1}{n})^{n}$
negative_passages.docid: 13525
negative_passages.text: Prove: If $n=2^k-1$, then $\binom{n}{i}$ is odd for $0\leq i\leq n$
negative_passages.docid: 136304
negative_passages.text: Prove: $2^b-1\mid{}2^a-1 \iff b\mid{}a$
negative_passages.docid: 39082
negative_passages.text: Proving $x^\alpha-\alpha x \le 1- \alpha $
negative_passages.docid: 7414
negative_passages.text: Prove that odd perfect square is congruent to $1$ modulo $8$
negative_passages.docid: 152039
negative_passages.text: Prove that $a^{2^n}=1 \mod 2^{n+2}$
negative_passages.docid: 5043
negative_passages.text: Show that $2^n>1+n\sqrt{2^{n-1}}$
negative_passages.docid: 150462
negative_passages.text: Prove by induction that for every $n \ge 0$ the term $2^{2n+1} - 9n^2 + 3n -2$ is divisible by $54$
negative_passages.docid: 8609
negative_passages.text: Prove by induction that $n!>2^n$