You are Here: Home >< Maths

# Integral coprime integer powers => integer ? Watch

1. What a title.

This is a small part of my solution to a larger question, if I can prove this then I prove the rest, and my brain has shut down. This is probably extremely easy.

Suppose x^a, x^b are integral for two integers a,b and gcd(a,b) = 1. Is x an integer?
2. No. Take a=b=-1 and x = 1/2.
3. Sorry a,b are positive coprime integers.
And another mistake, x^a and x^b are integers but not necessarily equal.
4. Yes.

Write and with primes.

By . This now implies for all . Since we must have and

Hence and

In other words,
5. If (a,b) = 1 then there exist m,n with am + bn = 1. So . So x is rational, and thus by a special case of the rational root theorem it must be an integer.

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: April 6, 2013
Today on TSR

### What is the latest you've left an assignment

And actually passed?

### Simply having a wonderful Christmas time...

Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.