Let $T=\underbrace{333\cdots 3}_{3^{2024}}$. Find the largest power of $3$ that can divid $T$.

Lifting The Exponent (LTE) Let $v_p(n)$ be the largest power of a prime $p$ that divides a positive integer $n$, and $x$, $y$ be any two integers such that $p \not\mid x$ and $p \not\mid y$, then

• When $p$ is odd
  • If $p\mid x-y$, then $v_p(x^n-y^n) = v_p(x-y)+v_p(n)$
  • If $n$ is odd and $p\mid x+y$, then $v_p(x^n+y^n) = v_p(x+y)+vp(n)$
  • If $n$ is even and $p\mid x+y$, then $v_p(x^n+y^n) = 0$
• When $p=$
  • If $2\mid x-y$ and $n$ is even, then $v_2(x^n-y^n)=v_2(x-y)+v_2(x+y)+v_2(n)-1$
  • If $2\mid x-y$ and $n$ is odd, then $v_2(x^n-y^n)=v_2(x-y)$
• For all $p$
  • if $gcd(n,p)=1$, and $p\mid x-y$, then $v_p(x^n-y^n)=v_p(x-y)$
  • If $bcd(n,p)=1$, $p\mid x+y$ and n odd, then $v_p(x^n+y^n) = v_p(x+y)$

Find all solutions in positive integers to $3^n = x^k + y^k$ where $x$ and $y$ are co-prime and $k\ge 2$.

Suppose $a$ and $b$ are both positive real numbers such as $a-b$, $a^2-b^2$, $a^3-b^3$, $\cdots$, are all positive integers. Show that $a$ and $b$ must be positive integers.