query: Number Theory-Find all natural numbers n such that
positive_passages.docid: 17243
positive_passages.text: Remainder when $2009^{2009}-1982^{2009}-1972^{2009}+1945^{2009}$ is divided by $1998$
negative_passages.docid: 115258
negative_passages.text: Find all positive integers $n$ such that $\phi(n)+\sigma(n)=2n$.
negative_passages.docid: 880
negative_passages.text: Find all n such that $\phi(n) = n/2$
negative_passages.docid: 105225
negative_passages.text: Number Theory: Complete set of residues modulo $n$
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: 108691
negative_passages.text: Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.
negative_passages.docid: 121694
negative_passages.text: How to prove $\log n \leq \sqrt n$ over natural numbers?
negative_passages.docid: 148906
negative_passages.text: Finding integers $m,n$ such that $(m-n)^2=4mn/(m+n-1)$
negative_passages.docid: 5375
negative_passages.text: Is there a bijective function mapping Natural numbers to Natural numbers, other than $f(n) = n$?
negative_passages.docid: 14783
negative_passages.text: A Number Theory problem (GCD)
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: 123427
negative_passages.text: Proving the sum of the first $n$ natural numbers by induction
negative_passages.docid: 3632
negative_passages.text: Basic number theory proofs
negative_passages.docid: 154475
negative_passages.text: Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number
negative_passages.docid: 13739
negative_passages.text: For all $n>2$ there exists a prime number between $n$ and $ n!$
negative_passages.docid: 93572
negative_passages.text: How to determine if $n (n \geq 2)$ given numbers can be summed up to a certain number?
negative_passages.docid: 126015
negative_passages.text: Elementary number theory in sets