Question

Consider the sequence of prime numbers denoted by P, where P = {2, 3, 5, 7, 11, 13, ...}. Let A be a subset of P containing 2019 distinct primes.

If the sum of all the elements in A is S, and each element in A is raised to the power of S, find the remainder when this resulting sum is divided by 1000.

Note: To clarify, if A = {2, 3, 5}, then the sum S = 2 + 3 + 5 = 10, and the resulting sum will be 2^10 + 3^10 + 5^10.