# A interesting thing about exponent... does this apply to everything?

Okay... when I was solving math questions today on Brilliant, one thing just jumped out of my mind.

It is easy to see 3^{3}-1 is divisible by 2. It is not hard to see 4^{4}-1 is divisible by 3. Is 5^{5}-1 also divisible by 4? Is 6^{6}-1 also divisible by 5?

Moreover, for example, is 17^{15}-1, 17^{16}-1, 17^{17}-1, 17^{18}-1 all divisible by 16?

I think it is. Here is my proof:

Let a and n be positive integers.

a^{n} =(a-1)(a^{n-1})+a^{n-1} =(a-1)(a^{n-1}+(a-1)(a^{n-2})-(a^{n-2}) =(a-1)(a^{n-1}+(a-1)(a^{n-2})-(a^{n-2})+......(a-1)a+(a-1)+1 after factoring, we get (a-1)(a^{n-1}+a^{n-2}+a^{n-3}+......+1)+1 therefore, a^{n}-1 =(a-1)(a^{n-1}+a^{n-2}+a^{n-3}+......+1)+1-1 =(a-1)(something)

and it is divisible by a-1.

But now I have another question. Does similar theory also apply to negative integers? (for example, let n be a negative integer)

Help me please! Thank you very much!

(P.S. I only learned Algebra 1 so sometimes I could be confused to use or even look at some notations. Thank you for your tolerance!)

Note by Margaret Zheng
3 years, 5 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. 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 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example 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 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

To check, do you know what Modular Arithemtic is? How about the Binomial Theorem?

Staff - 3 years, 5 months ago

I know binomial theorem, but I'm not sure with modular arithmetic. And thank you very much!

- 3 years, 5 months ago