Find the sum of all primes q<1000 such that for some prime p<q , both q divides p^3-1 and p divides q^3-1
To find the sum of all primes q less than 1000 that satisfy the given conditions, we need to check each prime number in that range and test if there exists another prime number p that satisfies the given conditions.
Let's break down the steps to solve this problem:
1. Create a function to check if a number is prime.
We can use a simple trial division algorithm to check if a number is prime. Iterate from 2 to the square root of the number, checking if any of those values divide the number evenly. If a factor is found, the number is not prime.
Here's an example implementation in Python:
```python
import math
def is_prime(n):
if n <= 1:
return False
for i in range(2, int(math.sqrt(n)) + 1):
if n % i == 0:
return False
return True
```
2. Iterate through the range of prime numbers less than 1000.
For each prime number, we will check if there exists another prime number p less than q that satisfies the given conditions.
```python
primes_sum = 0
for q in range(2, 1000):
if is_prime(q):
for p in range(2, q):
if is_prime(p) and q % p == 0 and p % q == 0:
primes_sum += q
break
```
In this code, if we find a prime number p that satisfies both conditions, we add q to the running sum and then break out of the inner loop.
3. Print the sum of all primes found.
Finally, we will print the sum of all primes q that satisfy the conditions.
```python
print(primes_sum)
```
Combining all the steps, here is the complete solution:
```python
import math
def is_prime(n):
if n <= 1:
return False
for i in range(2, int(math.sqrt(n)) + 1):
if n % i == 0:
return False
return True
primes_sum = 0
for q in range(2, 1000):
if is_prime(q):
for p in range(2, q):
if is_prime(p) and q % p == 0 and p % q == 0:
primes_sum += q
break
print(primes_sum)
```
Running this code will give you the sum of all primes q less than 1000 that satisfy the given conditions.