Is This The Hardest Prime Number Olympiad Problem Ever?

Описание: This problem is a masterclass in mathematical misdirection. At first glance, the claim that this product, Q(p), is an integer for ALL prime numbers seems absurd. The exponents turn negative, which should create a cascade of fractions. How could it possibly resolve into a perfect integer every single time?

This is a problem worthy of the highest levels of competition, like the Math Olympiad. Solving it requires moving beyond simple algebra and into the elegant world of number theory. In this video, we'll dissect the proof and uncover the beautiful logic hiding beneath the surface.

Join us as we explore:
► The Initial Sanity Check: Why the claim surprisingly holds for p=2, 3, and 5.
► The Core Strategy: Using p-adic valuations to analyze the prime factorization of Q(p).
► The Unexpected Tool: How a discrete version of "integration by parts" (Summation by Parts) transforms the problem.
► The Endgame: A stunning proof by contradiction using Legendre's Formula and the very definition of a prime number.

By the end, you'll not only see the solution but also understand why the primality of 'p' is the absolute key to the entire puzzle. This is one of those proofs that will change how you look at prime numbers.

Timestamps:
00:00 - The "Impossible" Problem
00:32 - Sanity Check: Testing Small Primes (2, 3, 5)
02:22 - The Strategy: Using Prime Factorization (p-adic Valuations)
04:13 - The Key Insight: Summation by Parts (Abel's Transformation)
07:22 - The Final Proof: Legendre's Formula and a Clever Inequality
11:17 - The "Aha!" Moment: Why Primality is CRUCIAL
12:58 - Conclusion: Tying It All Together
13:44 - Bonus: An Elegant Factorial Identity
14:29 - Why This Fails for Composite Numbers (Counterexample)