Prove that there are no positive integers $a, b$ and $n >1$ such that $a^n – b^n$ divides $ a^n + b^n$.
12694
Why does $a^n - b^n$ never divide $a^n + b^n$?
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 $m|n$?
96268
Prove that $5$ divides $3^{3n+1}+2^{n+1}$
69902
$3$ never divides $n^2+1$
158844
If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$
71650
Proving by strong induction for a sequence of integers, $2^n$ divides term $n$
4077
Using induction prove $n^3-n$ is divisible by 3 whenever n is a positive number.
63661
integers $n$ such that $(n-1)!+1$ is a power of $n$
80144
Prove that $x-1$ divides $x^n-1$
140405
Show that there does not exist any $n$ for which $a^n=a$ forall $a\in \Bbb Z_m$ where $m$ is divisible by the square of some prime.
82953
In a ring with no zero-divisors, for $(m,n) =1$, $a^m = b^m$ and $a^n = b^n$ $\iff a =b$
141102
If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.
110256
Let $a,b \in {\mathbb{Z_+}}$ such that $a|b^2, b^3|a^4, a^5|b^6, b^7|a^8 \cdots$, Prove $a=b$
75166
Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
83662
Prove that $ a $ divides $1^{a^{n}} + 2^{a^{n}} + ... + (a-1)^{a^{n}}$?
41784
Prove using mathematical induction that $n < 3 ^n$ for all positive integers n.
112631
Prove that if $d$ divides $n$, then $2^d -1$ divides $2^n -1$