Turn on thread page Beta
    • Thread Starter
    Online

    18
    ReputationRep:
    I was solving a problem. Stumbled upon a totally irrelevant problem. Found an awesome proof for it. The proof is very similar to another famous proof (won't say which one). Try it!

    Prove that there don't exist any non-integer rational numbers a,b such that a-b, a+b, ab are all integers.

    EDIT: apparently an easier proof exists too
    Offline

    22
    ReputationRep:
    Assume that a,b are non-integer rationals with a-b, a+b, ab all integer. Then p=2a and q=2b are integers and have to be odd. So pq is odd. But we know that pq/4 is an integer so pq is divisible by four. Contradiction.
    • Thread Starter
    Online

    18
    ReputationRep:
    (Original post by Zacken)
    Assume that a,b are non-integer rationals with a-b, a+b, ab all integer. Then p=2a and q=2b are integers and have to be odd. So pq is odd. But we know that pq/4 is an integer so pq is divisible by four. Contradiction.
    Yea. I found a much more indirect proof which was similar to proving \sqrt{2} is irrational. Basically assume a,b are rationals in simplest form and then arrive at contradiction.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: July 25, 2017
Poll
Favourite type of bread
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

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

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