** Prove that if for a number x ∈ R x + x ^(-1) ∈ Z , then for all n ∈ Z cases x^n + x^(−n) ∈ Z.strong text**
Asked
Active
Viewed 37 times
-2
-
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Nov 30 '20 at 07:33
-
Hi, welcome to MSE. Are you sure that is true? $$4 \in \mathbb{R}, 4^1+4^{-1}=\frac{17}{4} \not \in \mathbb{Z}$$ – Khosrotash Nov 30 '20 at 07:38
-
1You need to prove this using some form of induction. – player3236 Nov 30 '20 at 07:54
-
@Khosrotash ohh it's still hard to understand – tom2563 Nov 30 '20 at 08:30
-
@player3236 too it's still hard to understand – tom2563 Nov 30 '20 at 08:32
2 Answers
1
Induction hint: The cases $n=0$ and $n=1$ are trivial.
Suppose $x^{n-1} + x^{1-n}$ and $x^{n-2} + x^{2-n}$ are integers.
Consider the product $(x + x^{-1})(x^{n-1} + x^{1-n})$, and show that $x^n + x^{-n}$ must be an integer.
player3236
- 16,413
-
+1 : After finding a kludgy solution associated with my first hint, I found a more elegant Fibonacci-like approach associated with my 2nd hint. Your approach is even more elegant. – user2661923 Nov 30 '20 at 12:21
0
Hint
If $x + \frac{1}{x} = a$, then solve for $x$ in terms of $a$.
Then, prove the following lemma:
If $a$ is an integer and $b,n$ are a positive integers, then
$$(a + \sqrt{b})^n + (a - \sqrt{b})^n ~\text{is an integer}.$$
Hint-2
$$\left[a \times \frac{a \pm \sqrt{a^2 - 4}}{2}\right] ~-~ 1 ~=~ \left[\frac{a \pm \sqrt{a^2 - 4}}{2}\right]^2.$$
user2661923
- 35,619
- 3
- 17
- 39
-
-
-
@tom2563 Just providing you with these hints approached violating mathSE protocol. To say more would cross the boundary. See these links:https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question#9960 https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question/27933#27933. Your query has already received two down votes because it did not meet the mathSE standard discussed in these links. You need to edit your query in accordance with these links. – user2661923 Nov 30 '20 at 08:41
-
-
-
@tom2563 You have to edit your query to show that you have made a legitimate attempt to solve for $x$ in terms of $a$, discover the relationship between the problem and the lemma, and then prove the lemma. You do not have to succeed. But you do have to edit your query to show a serious attempt. – user2661923 Nov 30 '20 at 08:51