TITLE: Find $b$ when $(2+\sqrt{3})^n=5042+b\sqrt{3}$
QUESTION [5 upvotes]: Find $b$ when $(2+\sqrt{3})^n=5042+b\sqrt{3}$
I saw that $n$ cannot be bigger than $11$ because then the rational part is going to be bigger than $5024$. But I don't know how to find the $n$ exactly and how to find out $b$ then. Any hints?
Note: $b$ and $n$ are integers.

REPLY [5 votes]: First note that
$(2+\sqrt{3})^n = a+b\sqrt{3}$
with $a,b$ both positive whole  for any positive whole number $n$. There is also a conjugate relation
$(2-\sqrt{3})^n = a-b\sqrt{3}$
and then we have the product of these two relations
$1 = a^2-3b^2$
For any positive whole number $n$ the product relation implies that $b$ is the largest whole number less than $a/\sqrt{3}$.  So punch that quotient into your calculator with the given value of $a$ and read off the whole-number part.