I am struggling to see the great revelation.
I am not a mathematician, but in the discrete domain of integers:
1) you have two functions, essentially we look for where the two 3d space surfaces intersect, if z is taken as the result for each equation.
2) the integers "are disctinct", so (0,0) and (1,1) are out, plus (2,2) (3,3) etc. Basically a whole linear diagonal in the instersection of both 3d spaces is excluded (why though, to what useful end?)
3) Starting points for the ranges is therefore (0,1) and (1,0)
4) 1^y is always 1, and x^0 is always 1 so there is a constant starting value of 1 both on both axis
5) but x^y will always be larger than y^x, for y>x and x^y > y^x
(prove this by taking the first derivative to get rate of change. Do you use Laplace z domain for discrete, instead of s for continuous?)
6) and the converse to 5.
So once you have found one solution, you know to stop looking, the two surfaces keep diverging from each other.
Why resort to the continuous domain to solve a problem n the discrete, is this even a valid approach?
eg Is there a formal proof that says integers strictly follow that same rules as the continuous domain, just as a subset? I'm interested.
Does this come under group theory, a set with an applied operation?
As I said I am not a mathematician, I am an electrical engineer so probably one of the worst abusers of pure math in a formal sense, but the more I think about this the more questions this raises in my thinking.
Can someone point out errors in thinking?
I am not a mathematician, but in the discrete domain of integers:
1) you have two functions, essentially we look for where the two 3d space surfaces intersect, if z is taken as the result for each equation.
2) the integers "are disctinct", so (0,0) and (1,1) are out, plus (2,2) (3,3) etc. Basically a whole linear diagonal in the instersection of both 3d spaces is excluded (why though, to what useful end?)
3) Starting points for the ranges is therefore (0,1) and (1,0)
4) 1^y is always 1, and x^0 is always 1 so there is a constant starting value of 1 both on both axis
5) but x^y will always be larger than y^x, for y>x and x^y > y^x (prove this by taking the first derivative to get rate of change. Do you use Laplace z domain for discrete, instead of s for continuous?)
6) and the converse to 5.
So once you have found one solution, you know to stop looking, the two surfaces keep diverging from each other.
Why resort to the continuous domain to solve a problem n the discrete, is this even a valid approach?
eg Is there a formal proof that says integers strictly follow that same rules as the continuous domain, just as a subset? I'm interested.
Does this come under group theory, a set with an applied operation?
As I said I am not a mathematician, I am an electrical engineer so probably one of the worst abusers of pure math in a formal sense, but the more I think about this the more questions this raises in my thinking.
Can someone point out errors in thinking?