I was hoping that some of you with a lot more experience than me would check my work below for any blatant errors or incorrect assumptions, I have marked the parts I'm unsure about. I have deliberately included every step for clarity.

Many thanks in advance.

Two equations:

......... (1)

............ (2)

Restrictions:

u and v are integers

a, b and c are integers a > b > c > 0

I am trying to prove, given the 2 equations and restrictions, that there are no valid values for a, b or c. Proof that any one of them can't be an integer within the restrictions is enough.

From (2)

manipulating the RHS (I'll call this Step 1 for a later question)

everything ok so far? (I think the min ratio of u:v has increased here, but not important at the minute)

from (3) ....

from (4) ....

manipulating LHS

from (6) ....

substituting this in (8) ...

for to be +ve integer must be a perfect square.

no manipulation here, just grouping...

are these two roots attained correctly?

So this proves that cannot have a value that fits within the given constraints.

Question about Step 1. Does what I have shown only prove there's no valid when the equations are manipulated as in Step 1, or does it prove it conclusively?

If not conclusively, there are infinite ways of manipulating the equation at that stage

If this is the case, is there any way to prove conclusively what I'm attempting?

Many thanks for your time