Cousin Primes & Digital Roots in Python

🔍 Cousin Primes & Their Digital Roots

🎯 What Are Cousin Primes?

Cousin primes are pairs of prime numbers that differ by exactly 4. Examples include (3, 7), (7, 11), and (13, 17). These pairs offer insights into prime gaps and distribution patterns.

💡 What Is a Digital Root?

The digital root of a number is the single-digit value obtained by repeatedly summing its digits until only one digit remains. For example:

• Digit root of 137 → 1 + 3 + 7 = 11 → 1 + 1 = 2
• Digit root of 89 → 8 + 9 = 17 → 1 + 7 = 8

💻 Python Code

def is_prime(n):
    if n < 2:
        return False
    for i in range(2, int(n**0.5) + 1):
        if n % i == 0:
            return False
    return True

def digit_root(n):
    while n >= 10:
        n = sum(int(d) for d in str(n))
    return n

def find_cousin_primes_with_roots(limit):
    cousin_pairs = []
    for p in range(2, limit - 4):
        if is_prime(p) and is_prime(p + 4):
            dr1 = digit_root(p)
            dr2 = digit_root(p + 4)
            cousin_pairs.append(((p, p + 4), (dr1, dr2)))
    return cousin_pairs

def main():
    try:
        user_limit = int(input("Enter the upper limit to find cousin primes: "))
        pairs_with_roots = find_cousin_primes_with_roots(user_limit)
        print(f"\nCousin Prime Pairs with Digital Roots up to {user_limit}:")
        for (p1, p2), (dr1, dr2) in pairs_with_roots:
            print(f"({p1}, {p2}) → ({dr1}, {dr2})")
    except ValueError:
        print("Please enter a valid integer.")

# Run the program
main()

Copy and Try it here!

📊 Sample Output

Input: 50

Output:

(3, 7) → (3, 7)
(7, 11) → (7, 2)
(13, 17) → (4, 8)
(19, 23) → (1, 5)
(37, 41) → (1, 5)
(43, 47) → (7, 2)

🔍 Why It’s Fascinating

Digital roots offer a compact way to analyze numerical behavior. When applied to cousin primes, they reveal patterns, symmetries, and attractors that might otherwise go unnoticed. This blend of number theory and digit analysis is perfect for curious minds.

🌟 Final Thoughts

Try different limits and observe how digital roots behave across cousin primes. Are certain root pairs more frequent? Do they repeat cyclically? This script is a great way to explore prime behavior and deepen your mathematical intuition.