Find the sum of all positive integers b such that b^2=a^3+1, where a is a prime number.
To find the sum of all positive integers b that satisfy the given equation, we can follow these steps:
1. Iterate over all prime numbers (a) greater than or equal to 2.
2. For each prime number (a), calculate a^3 + 1.
3. Check if the result of step 2 is a perfect square (b^2) by using a square root function.
4. If the result is a perfect square, add the positive square root (b) to a running sum.
5. Repeat steps 2-4 for all prime numbers.
6. Return the final sum.
Here is a Python code implementation of the above steps:
```python
import math
def is_perfect_square(num):
return math.isqrt(num) ** 2 == num
def find_sum_of_b():
prime_sum = 0
for a in range(2, 1000000):
if all(a % i != 0 for i in range(2, int(math.sqrt(a)) + 1)):
cube_plus_one = a ** 3 + 1
if is_perfect_square(cube_plus_one):
prime_sum += math.isqrt(cube_plus_one)
return prime_sum
sum_of_b = find_sum_of_b()
print(sum_of_b)
```
By running this code, you will obtain the sum of all positive integers (b) that satisfy the given equation.