Can You Prove This Factorial Expression is ALWAYS Even?

Описание: Dive deep into a fascinating number theory problem that challenges us to prove that the floor of (n-1)! / (n^2 + n) is always an even number for any positive integer 'n'. This video provides a complete, step-by-step proof that is both rigorous and easy to follow.
We start by simplifying the expression and testing small values of 'n' to build intuition. Then, we tackle the core of the proof for n ≥ 6 using a powerful and elegant method: comparing the 2-adic valuations of the numerator and the denominator. You'll learn how to apply Legendre's Formula to find a strong lower bound for the number of factors of two in (n-1)! and how to establish a logarithmic upper bound for the factors of two in the denominator.
Join us as we:
Prove a crucial lemma about composite numbers and factorials.
Use Legendre's Formula to analyze prime factorizations.
Rigorously prove that a linear growth function will always outpace a logarithmic one.
Show why this elegant 2-adic argument provides a uniform proof that covers all cases, including when 'n' or 'n+1' are prime.
Whether you're a math student looking to deepen your understanding of number theory, or just someone who loves a good mathematical puzzle, this video will guide you through a beautiful and satisfying proof.

Video Timestamps
0:00 - The Problem: Proving floor((n-1)! / (n^2 + n)) is Always Even
0:19 - Step 1: Simplification and Direct Verification
0:28 - Simplifying the Denominator
0:53 - Testing Small Values of n (n=1 to 5)
1:25 - Step 2: Formal Analysis for n ≥ 6
1:39 - Illustrative Case: When n and n+1 are Composite
1:50 - Proving a Key Lemma: m | (m-1)! for Composite m ≥ 6
3:01 - Applying the Lemma to Prove Integrality
4:02 - A More Powerful Method: The Uniform 2-adic Proof for All n ≥ 6
4:14 - Strategy: Comparing Factors of Two (v₂(A) ≥ 1)
4:38 - Numerator Analysis with Legendre's Formula
4:50 - Denominator Analysis and its Logarithmic Bound
5:14 - Strengthening the Bound for the Numerator
5:35 - Proving the Inequality by Comparing Growth Rates
6:23 - The Rigorous Conclusion for n ≥ 6
6:56 - Final Conclusion: Summarizing the Complete Proof