On the Uniqueness of Prime Solutions to the Equation $a^n+b^n=n^c+n^d$

We study the structure of solutions to the exponential Diophantine equation$$a^n+b^n=n^c+n^d,\qquad a,b,c,d,n\in\N$$under the condition that all variables are prime numbers. Through elementary number-theoretic analysis and modular constraints, we prove that if $a,b,c,d,n$ are all prime and satisfy the above equation, then necessarily $a=b=c=d=n$. This result shows that the prime constraint reduces the solution space of the equation from an infinite family to the diagonal form $(p,p,p,p,p)$. The paper also provides computational verification for $n<100$ and discusses possible directions for generalization.