Prime Constellations & Digital Roots

🌌 Prime Constellations & Their Digital Roots

🎯 What Are Prime Constellations?

Prime constellations are structured patterns of prime numbers separated by fixed gaps. Examples include:

• Twin Primes: (p, p+2)
• Cousin Primes: (p, p+4)
• Sexy Primes: (p, p+6)
• Triplets: (p, p+2, p+6) or (p, p+4, p+6)
• Quadruplets: (p, p+2, p+6, p+8)

💡 What Is a Digital Root?

The digital root of a number is the single-digit result of repeatedly summing its digits. It helps reveal hidden numerical patterns and attractors.

💻 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 generate_constellations(lower, upper, gaps):
    results = []
    for p in range(lower, upper - max(gaps)):
        if all(is_prime(p + g) for g in gaps):
            drs = [digit_root(p + g) for g in gaps]
            results.append(((p,) + tuple(p + g for g in gaps), tuple(drs)))
    return results

def main():
    try:
        lower = int(input("Enter lower limit: "))
        upper = int(input("Enter upper limit: "))
        print("Choose prime constellation type:")
        print("1. Twin (p, p+2)")
        print("2. Cousin (p, p+4)")
        print("3. Sexy (p, p+6)")
        print("4. Triplet (p, p+2, p+6)")
        print("5. Triplet (p, p+4, p+6)")
        print("6. Quad (p, p+2, p+6, p+8)")
        choice = int(input("Enter choice (1–6): "))

        gap_map = {
            1: [2],
            2: [4],
            3: [6],
            4: [2, 6],
            5: [4, 6],
            6: [2, 6, 8]
        }

        selected_gaps = gap_map.get(choice)
        if not selected_gaps:
            print("Invalid choice.")
            return

        results = generate_constellations(lower, upper, selected_gaps)
        print(f"\nPrime Constellations with Digital Roots from {lower} to {upper}:")
        for primes, drs in results:
            print(f"{primes} → {drs}")

    except ValueError:
        print("Please enter valid integers.")

main()

Copy and Try it here!

📊 Sample Output

Input: Lower = 10, Upper = 50, Choice = 4 (Triplet: p, p+2, p+6)

Output:

(11, 13, 17) → (2, 4, 8)
(17, 19, 23) → (8, 1, 5)
(31, 33, 37) → (4, 6, 1)

🔍 Why It’s Powerful

This tool lets you explore prime constellations interactively and analyze their digital roots. It’s perfect for discovering attractor patterns, testing hypotheses, and building intuition for prime gaps.

🌟 Final Thoughts

Try different ranges and constellation types. Are certain digital root combinations more frequent? Do they repeat cyclically? This script is a gateway to deeper number theory exploration and a great addition to your blog series.