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

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2
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2answers
55 views
+50

What's the formula for the fixed point of a power tower of $2$s modulo $n$?

e.g., $n = 3$. Clearly, the powers of $2$ modulo $3$ alternate $2, 1$. The powers of $4$ modulo $3$ are all $1$s. So a power tower of $2$s modulo $3$ must get stuck on $1$. Of course this method is ...
7
votes
3answers
376 views
+50

How does one attack a divisibility problem like $(a+b)^2 \mid (2a^3+6a^2b+1)$?

In my current line of investigation, I am running into [many] divisibility questions like the one in the title, i.e. $$ (a+b)^2 \mid (2a^3+6a^2b+1), \qquad(\star) $$ where $a > b \ge 1$ are ...