Roots of Identity Matrix

I was reading through some problems based on permutations. Particularly, the following problem interested me.

Let Zn\mathbb{Z}_n denote the set of integers {1,2,,n}\{1,2,\ldots,n\} for any positive integer nn and let f:ZnZnf : \mathbb{Z}_n \rightarrow \mathbb{Z}_n be a bijective function defined on Zn\mathbb{Z}_n.

Given that nn is finite, does integer kk^* exist, such that it is always possible to find a kkk \le k^* for which fk(x)=x,xZnf^k(x)=x, \forall x \in \mathbb{Z}_n?

I suspect that this would turn out to finding the primitive real roots of the identity matrix. That is finding the maximum value of kk such that Xk=InX^k=I_n has a n×nn \times n matrix solution XX , but, XiInX^i \neq I_n for any i<ki < k

Taking into account that any arbitrary permutations could consist of mm cycles of length i1,i2,,imi_1,i_2,\cdots,i_m with i1+i2++im=ni_1+i_2+\cdots+i_m=n. The problem now boils down to finding the maximum value of the LCM of i1,i2,,imi_1,i_2,\cdots,i_m over all possible values of mm i..e,

k=max\substackm=1n,i1+i2++im=nLCM(i1,i2,,im) k^* = \max_{\substack{m=1\cdots n,\\ i_1+i_2+\cdots+i_m=n}} LCM (i_1,i_2,\cdots,i_m) But, am at a loss in actually proceeding much further.

Note by Janardhanan Sivaramakrishnan
5 years, 1 month ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


There are no comments in this discussion.


Problem Loading...

Note Loading...

Set Loading...