I think this is interesting !

[Commander]

Now and then I play around with Maths and Scientific thinking.

 

Only later on I check on the available knowledge store.

 

Two days back I thought of testing out the following equation :

 

 

x ^y = y ^x where both x and y are Positive Integers and x is not equal to y.

 

I guessed that there won't be a solution to this equation then I realized that

 

2 ⁴ = 4 ² and perhaps this is the only solution possible.

 

Are there any other solutions ?

[ajb]

You can prove that 2 and 4 are the only integer solutions.

 

One method is to look for rational solutions and from there you can deduce that there are no other integer solutions. You can find proofs of this via a quick google. See for example http://www.qbyte.org/puzzles/p048s.html

[Commander]

You can prove that 2 and 4 are the only integer solutions.

 

One method is to look for rational solutions and from there you can deduce that there are no other integer solutions. You can find proofs of this via a quick google. See for example http://www.qbyte.org/puzzles/p048s.html

 

Hi Thanks,

 

I also note that if we take the adjacent pairs of integers ;

 

1² < 2¹

2³ < 3²

 

but thereafter

 

3⁴ > 4³

4⁵ > 5⁴

5⁶ > 6⁵

 

etc and this nⁿ⁺¹ > (n+1)ⁿ will hold for higher number adjacent pairs.

[ajb]

... nⁿ⁺¹ > (n+1)ⁿ will hold for higher number adjacent pairs.

You can prove this first by simplifying the inequality and then by induction. Maybe for fun you can work this out and post a proof here?