Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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Why it is impossible for primitive Pythagoras triplets in integers to be all as powerful numbers?

I had seen an elementary proof for Fermat's last theorem at Quora. I had checked all the steps (around one page only),where I couldn't catch any error, but I was confused about the last step only ...
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How to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]?

$\forall \alpha\in [0,1]\setminus\mathbb{Q}$, how to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]? $\{x\}$ is the fractional part of x. Any hint would be appreciated!
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Prove or disprove that $2^n$ divides $T_{2^n}$ for $n > 2$.

The Tribonacci sequence satisfies $$T_0 = T_1 = 0, T_2 = 1,$$ $$T_n = T_{n-1} + T_{n-2} + T_{n-3}.$$ Prove or disprove that $2^n$ divides $T_{2^n}$ for $n > 2$. (I think $2^n$ divides $T_{2^n}$...