Sexy Primes & Digital Roots in Python

💫 Sexy Primes & Their Digital Roots

🎯 What Are Sexy Primes?

Sexy primes are pairs of prime numbers that differ by exactly 6. The name comes from the Latin word “sex” meaning six—not from anything risqué! Examples include (5, 11), (7, 13), and (11, 17).

💡 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:

• 137 → 1 + 3 + 7 = 11 → 1 + 1 = 2
• 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_sexy_primes_with_roots(lower, upper):
    sexy_pairs = []
    for p in range(lower, upper - 6):
        if is_prime(p) and is_prime(p + 6):
            dr1 = digit_root(p)
            dr2 = digit_root(p + 6)
            sexy_pairs.append(((p, p + 6), (dr1, dr2)))
    return sexy_pairs

def main():
    try:
        lower_limit = int(input("Enter the lower limit: "))
        upper_limit = int(input("Enter the upper limit: "))
        if lower_limit >= upper_limit:
            print("Lower limit must be less than upper limit.")
            return
        pairs_with_roots = find_sexy_primes_with_roots(lower_limit, upper_limit)
        print(f"\nSexy Prime Pairs with Digital Roots from {lower_limit} to {upper_limit}:")
        for (p1, p2), (dr1, dr2) in pairs_with_roots:
            print(f"({p1}, {p2}) → ({dr1}, {dr2})")
    except ValueError:
        print("Please enter valid integers.")

# Run the program
main()

Copy and Try it here!

📊 Sample Output

Input: Lower = 10, Upper = 50

Output:

(11, 17) → (2, 8)
(17, 23) → (8, 5)
(23, 29) → (5, 2)
(31, 37) → (4, 1)
(37, 43) → (1, 7)

🔍 Why It’s Fascinating

Digital roots offer a compact way to analyze numerical behavior. When applied to sexy 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 ranges and observe how digital roots behave across sexy 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.