Prove k is a Multiple of 6 When p and (p+2) are Prime and p + (p+2) = k^2

Problem Introduction

• 💡 The video addresses a mathematical proof: if p and p+2 are prime numbers (twin primes) and their sum, p + (p+2), equals k² for some integer k, then k must be a multiple of 6.
• 📌 This problem was inspired by an article by Vipra Raja on Medium, which explores conditions for the sum of twin primes to be a perfect square.

Initial Examples and Simplification

• 🎯 The proof starts with examples: for twin primes (71, 73), their sum 2(71)+2 = 144, which is 12² (so k=12). For (881, 883), their sum 2(881)+2 = 1764, which is 42² (so k=42). Both 12 and 42 are multiples of 6.
• 🧩 The problem statement p + (p+2) = k^2 simplifies to 2p + 2 = k², or 2(p+1) = k².

Proving Divisibility by Two

• ✅ Since 2(p+1) = k², the left side is clearly an even number.
• 🔑 This implies that k² must be even, which further means that k itself must be an even number.
• 💡 Therefore, 2 divides k, establishing the first part of the proof without complex results.

Proving Divisibility by Three

• 🔬 The proof then focuses on showing that 3 divides k.
• 🧠 A key property used is that any integer k² when divided by 3 can only have a remainder of 0 or 1 (never 2).
• ⚠️ Assuming k is not a multiple of 3, then k² must be congruent to 1 (mod 3).
• 🔄 Substituting k² = 2(p+1) into this congruence gives 2(p+1) ≡ 1 (mod 3).
• 🧩 This congruence simplifies to p ≡ 1 (mod 3).
• 🚨 If p ≡ 1 (mod 3), then p+2 ≡ 0 (mod 3), meaning p+2 is divisible by 3.
• 🛑 Since p+2 is a prime number and is divisible by 3, it must be that p+2 = 3.
• ❌ This leads to p = 1, which is not a prime number, creating a contradiction.
• 📈 The contradiction proves that the initial assumption (k is not a multiple of 3) is false, so k must be a multiple of 3.

Conclusion of the Proof

• 🏆 With k proven to be a multiple of both 2 and 3, and since 2 and 3 are coprime, it logically follows that k must be a multiple of 6.
• 💬 The video notes that the question of whether there's an infinite number of twin primes remains an unproven conjecture, which is related to the number of solutions for this problem.