Let 𝑎,𝑏,𝑚 and 𝑛 be natural numbers , 𝑎>1, and suppose that 𝑎,𝑏 have no common factor. Prove that $(𝑎^𝑛 + 𝑏^𝑛)|(𝑎^𝑚+𝑏^𝑚)\implies 𝑛|𝑚 $
70006
$a,b,m,n$ be positive integers such that g.c.d.$(a,b)=1$ and $a^m+b^m|a^n+b^n$ , then $m|n$?
95111
Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
131754
Prove $ \forall c \in \mathbb{R^{+}}, \forall B \in \mathbb{N}, \exists n \in \mathbb{N}, (n \geq B) \wedge (\sqrt{2n} > c \ln(n+2)) $?
95889
Prove that $\gcd(n! + 1, 3n! + 1) = 1$, $\forall n \in \mathbb{N}$, $n \geq 2$.
89175
Proof of $(a+b)^{n+1}$
55266
Prove that $(n+1)^{n-1}<n^n$
156766
Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
149716
Prove that $ \sqrt{n(n+1)(n+2)(n+3)+1} $ is always a natural number
84417
Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$
100375
Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}}$.
95061
Proving that $2^{2\cdot 3^{n-1}}\equiv 1+3^n\pmod{3^{n+1}}$ for every natural $n$
98855
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
145519
Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$
124320
$a,b,c>0 \text{ s.t. }a+b+c=1 \implies \sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ca+b} \ge 1+ \sqrt{ab}+\sqrt{bc}+\sqrt{ca}$
25648
Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$
109305
Non-existence of natural numbers such that $\sqrt{n} +\sqrt{n+1} <\sqrt{x} +\sqrt{y} <\sqrt{4n+2}$
127843
Prove that, for any natural number $k$ $\exists m \in M $ s.t i) $m$ has exactly $k$ $1$'s, an