To solve this problem, we need to find all primes p less than 1000 for which the number of polynomials satisfying the given condition is strictly greater than p * 2^(p-2). Let's break down the steps to solve this problem:
1. Generate a list of primes less than 1000:
- To do this, you can use various prime number generation algorithms, such as the Sieve of Eratosthenes or the Miller-Rabin primality test. Implement one of these algorithms to generate a list of primes.
2. Check each prime p in the list:
- For each prime p, we need to count the number of polynomials satisfying the given condition.
3. Count the number of polynomials satisfying the condition:
- Iterate through all possible polynomials f(x) with integer coefficients from 1 to p and degree at most p-1.
- For each polynomial, check whether f(2x) - f(x) is divisible by p for all integers x.
- Keep track of the count of polynomials that satisfy this condition.
4. Compare the count with p * 2^(p-2):
- Once the count of polynomials satisfying the condition is obtained, compare it with p * 2^(p-2).
- If the count is strictly greater than p * 2^(p-2), add p to the sum.
5. Calculate the sum:
- Sum up all the primes p for which the count is strictly greater than p * 2^(p-2).
- The result will be the sum of all such primes.
Implementing the above steps will allow you to find the sum of primes satisfying the given condition.