+5 Let P be the smallest prime such that there exist positive integers A and B satisfying \(A^2+P^3=B^4\)
Find all possible values of A.
+140 We can approach this problem by systematically checking prime numbers and their corresponding values of A and B.
1. Small Prime Numbers:
For P = 2:
We need to find A and B such that A^2 + 2^3 = B^4.
This simplifies to A^2 + 8 = B^4.
By trial and error, we find that A = 1 and B = 3 satisfy this equation.
2. Larger Prime Numbers:
As P increases, the value of B^4 grows much faster than A^2 + P^3.
This means that for larger primes, it becomes increasingly difficult to find integer solutions for A and B.
In fact, it can be shown that there are no other solutions for larger prime numbers.
Therefore, the only possible value of A is 1, corresponding to the prime number P = 2.
