Let, $\Bbb{P}$ denote the set of all odd prime numbers and $\Bbb{N}$ be the set of all natural numbers. Let, $2a,2b$ be two even numbers both greater than $4$.
Define, $A=\{(p,q)\in\Bbb{P}\times\Bbb{P}:p+q=2a\}$ and $B=\{(r,s)\in\Bbb{P}\times\Bbb{P}:r+s=2b\}$. Let, $X'=\{A,B\}$
Define, $n:X'\to \Bbb{N}$ by, $$n(Y)=\max_{(x,y)\in Y}\{x^2+y^2\},\forall Y\in X'=\{A,B\}$$ My claim is, $n(A)=n(B)\implies A=B$
I have no proof as well as no counterexample for my claim. If anyone has any idea about how to prove this or have seen any paper on this or it is very trivial to prove or has a counterexample for this claim please give.
Thanks in advance!
P.S.: I am "in danger of being blocked from asking any more".
I'm afraid the question is not well defined; or at least, not proven to be well defined. In your definition for the sets $A$ and $B$, you essentially say they are the sets of all pairs of odd primes which sum to $2a$ and $2b$, which are even numbers. The Goldbach Conjecture asserts that every even number ($>4$) can be expressed as the sum of two odd primes, and this unfortunately hasn't been proven. Hence, if the Goldbach Conjecture happens to be false, your sets $A$, $B$ become null sets, and so the question becomes ill-defined and doesn't make sense; indeed, asserting the well-definition of $A$, $B$ would be asserting Goldbach! (This is of course unless you're willing to define $\max(x^2+y^2)$ to mean something else in this case, in which case your conjecture comes down to asserting that there is exactly one even positive integer that can't be expressed as a sum of odd primes, an equally tough problem to prove.)