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Conjugate Numbers

Post Icon Posted: Submitted by richardhp on 26 June 2008 - 10:41am.

Joined: 2007-10-01
Posts: 170

I was reading about Young's Inequality where it says something about "conjugate numbers" being $ p,q >1\, $ satisfying $ \, \frac{1}{q} + \frac{1}{p} = 1 $

But I could only think of the number 2 satisfying this equation. How would you go about solving this in general?

‹ support of a function What's so special about Cauchy sequences ›
Post Icon Posted: 26 June 2008 - 12:17pm

Joined: 2006-11-02
Posts: 1019
If $ p,q $ are integers, the only solution is indeed 2:
$$\frac{1}{q} + \frac{1}{p} = 1 \quad \Longrightarrow \quad q + p = qp \quad \Longrightarrow \quad q = p(q-1) \quad \Longrightarrow \quad q-1 \mid q \quad \Longrightarrow \quad q = 2.$$

If you're looking for solutions in the reals though (as in the case of Young's inequality), you can find a $ p $ for any $ q>1 $ by using the fact that $ \frac{1}{p}=1-\frac{1}{q}=\frac{q-1}{q} $ and hence $ p = \frac{q}{q-1} > 1 $ works. Using this you can get a lot of solutions; for example $ \frac{1}{\pi} + \frac{1}{\pi(\pi-1)^{-1}} = 1. $
Post Icon Posted: 26 June 2008 - 12:28pm

Joined: 2006-10-10
Posts: 519

You just gotta fiddle with the form of the p's and q's.

So, for example, 1/4 + 3/4 = 1. Then, p = 4, and q = 4/3.

2,2 is the only integer solution though, I'm guessing.

EDIT: Psh, my solution was clearly superior. Would you like to borrow my Inequalities book over the summer? If you're planning to read Measure Theory, it should be a real help
Post Icon Posted: 26 June 2008 - 1:28pm

Joined: 2007-10-01
Posts: 170

ok so i was just being stupid then.
Post Icon Posted: 26 June 2008 - 1:51pm

Joined: 2006-10-10
Posts: 519

A guess, but were you trying to think of the problem as finding two numbers that have the same sum and product? That's how I tried to solve it at first and got myself really confused. Then I just thought of the original question and it came apparent. Probably a moral there somewhere
Post Icon Posted: 27 June 2008 - 11:45am

Joined: 2007-10-01
Posts: 170

yeah, don't do drugs kids.
Post Icon Posted: 27 June 2008 - 2:03pm

Joined: 2006-10-10
Posts: 519

Does anyone remember doing this as a STEP question? It was in that book of past questions
Post Icon Posted: 27 June 2008 - 3:41pm

Joined: 2006-11-02
Posts: 1019

Yes, actually, now that you mention it.

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