Series of trial-and-error questions:
Find all the unknowns for:
1. x^y=y^x
2. x^y=y^x+1
where all the unknowns are integers.

Question

lollylau · Answer

?????????!!!!!!!!

anonymous · Answer

i guess 
$$y=x$$ might work for first one

lollylau · Answer

could be...

lollylau · Answer

let's say they aren't, some actual answers please...

anonymous · Answer

what is wrong with 
$$3^3=3^3$$?

anonymous · Answer

2 and 4 works for the first one.

lollylau · Answer

Nothing!
Pratu got it right :)
(Well you too...)

mr.math · Answer

Sorry, I thought it was a system of two equations.

lollylau · Answer

oh.

anonymous · Answer

ooooooooooooooh. i was wondering...

mr.math · Answer

I agree with satellite on the first one, the solution of the first one is all integers (x,y) such that x=y.
For the second one, we have (x,y)=(2,1) is a solution.

mr.math · Answer

(3,2) is another solution.

anonymous · Answer

pratu had 
$$2^4=4^2$$ so that works as well. my guess is there are no others, but that is a guess

lollylau · Answer

There are no others.

lollylau · Answer

no proof tho. just know that for SURE :)

lollylau · Answer

Mr. Math, you're missing one more answer. (which i know)

lollylau · Answer

(solution)

lollylau · Answer

ans. in half a min.

mr.math · Answer

(1,0).

lollylau · Answer

YAY!

mr.math · Answer

Actually any solution in the form (x,0), x is a positive integer.

lollylau · Answer

Can we do this for +2, +3, +4 etc.

mr.math · Answer

Yes.