Why Is This Weird Expression Always Divisible by 4? | #discretemath | #numerology | #dogmathic

#Dogmathic #NumberTheory #DiscreteMath #Bijection #MathProof This Expression Is Divisible by 4. Then It Gets Weird. Divisibility and bijection don't usually show up in the same five minutes, but this video gets there. We start by proving that 1 + (−1)ⁿ(2n−1) is divisible by 4 for every positive integer n. Not by induction this time. Cases work better here. Split it into n odd and n even. In the odd case, n becomes 2k+1, the power of −1 collapses to −1 every time, and the algebra reduces to 4k. In the even case, n becomes 2k, the power of −1 collapses to positive 1 every time, and the algebra reduces to 4k again. Same answer from two different directions. That's not nothing. Once the divisibility is settled, the real payoff shows up. Define f(n) as the expression divided by 4 and start plugging in n = 1, 2, 3, 4, 5. The outputs go 0, 1, negative 1, 2, negative 2, and keep alternating like that forever. A function from the positive integers landing on every single integer, each one exactly once. That's a bijection, built out of a divisibility proof nobody asked for. There's a small challenge buried in here too. Proving the function is bijective means showing it's both one to one and onto, and that's left as an exercise for anyone who wants to actually do it instead of just nodding along. Topics covered: divisibility proof, bijection, case proof, odd and even integers, one to one function, onto function, integer mapping, discrete math, number theory, proof techniques, divisibility by 4, function definition, positive integers, set of integers Support Dogmathic https://ko-fi.com/dogmathic https://dogmathic.com/ matherssen(at)gmail.com    • My Handwriting Is Bad but This Proof Is Be...      • Ruby!      • Induction Proofs      • Discrete Mathematics      • Summations      • Number Theory   Properties and Concepts Used: Proof by cases Definition of odd integers (n = 2k+1) Definition of even integers (n = 2k) Properties of powers of negative one Divisibility (definition and proof) Algebraic simplification and distribution Function definition (f: ℤ⁺ → ℤ) One-to-one (injective) functions Onto (surjective) functions Bijection Pattern recognition in sequences Integer mapping Chapters: 0:00 Setting up the problem 0:29 Why cases instead of induction 1:12 Case 1: n is odd 3:19 Simplifying the odd case to 4k 4:05 Case 2: n is even 5:14 Simplifying the even case to 4k 6:30 Why this divisibility result is interesting 7:24 Defining f(n) and testing values 9:41 The pattern emerges: a bijection from ℤ⁺ to ℤ 10:23 Challenge: prove it's bijective #Dogmathic #NumberTheory #DiscreteMath #Bijection #MathProof