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Bring it in compa #runninglifestyle#motivation#Fitness#healthyliving,#Running#fyp#runner#runtok#run
Bring it in compa #runninglifestyle#motivation#Fitness#healthyliving,#Running#fyp#runner#runtok#run
The American dream🇺🇸🦅 Four brilliant Actors give away hugs.🥰 #truecringecomunnity #fyp #viral #larp #tfd Graham’s number is one of the most extreme examples of a finite but unimaginably large number ever to appear in a legitimate mathematical proof. It was introduced by mathematician Ronald Graham in 1971 (joint work with Bruce Rothschild) and was popularized by Martin Gardner in his Scientific American column. The Mathematical Problem Behind It Graham’s number arises in Ramsey theory, which studies conditions under which order must appear in large enough structures, even if things look random. Specifically, the problem is about coloring the edges of a high-dimensional hypercube: • Take an n-dimensional hypercube (a generalization of a square in 2D or a cube in 3D). • Every pair of vertices (corners) is connected by an edge. • Color every edge either red or blue. The question: What is the smallest dimension n such that no matter how you color the edges, you are guaranteed to have a set of 4 coplanar vertices that form a complete graph (a “square” or K₄) where all edges are the same color (all red or all blue)? This is a Ramsey-type question: forcing monochromatic structures in large enough systems. • We know such an n exists (by Ramsey theory arguments). • The best known lower bound is relatively small (something like 13 or so — people have found colorings that avoid monochromatic planar K₄ up to certain dimensions). • Graham’s number was an upper bound: it proves that the monochromatic planar K₄ is guaranteed to appear by dimension Graham’s number (or much earlier). In other words, Graham’s number is a (very loose) upper bound for this Ramsey number. The actual value is almost certainly tiny compared to Graham’s number — probably under 100 or even much smaller — but proving tight bounds in Ramsey theory is notoriously hard. Why Is It So Big? The size comes from the recursive way it’s constructed using Knuth’s up-arrow notation, which extends ordinary arithmetic operations in a tower of increasingly powerful operations. Quick Refresher on Up-Arrow Notation • a ↑ b = a^b (exponentiation)
Example: 3 ↑ 3 = 27 • a ↑↑ b = a^(a^(a^…)) with b many a’s — a power tower of height b (tetration)
Example: 3 ↑↑ 3 = 3^(3^3) = 3^27 = 7,625,597,484,987 • a ↑↑↑ b = a ↑↑ (a ↑↑ (a ↑↑ …)) with b many a’s — a “tetration tower” evaluated from the top down or right-associatively. • a ↑↑↑↑ b = a ↑↑↑ (a ↑↑↑ … ) with b many a’s — and so on. Each additional arrow creates a massive jump in growth rate. Exact Definition of Graham’s Number Graham’s number is usually denoted as g₆₄ and is defined in 64 recursive steps: • Let g₁ = 3 ↑↑↑↑ 3
(3 with four up-arrows to 3) • g₂ = 3 ↑^{g₁} 3
(3 with g₁ many up-arrows to 3) • g₃ = 3 ↑^{g₂} 3
(3 with g₂ many up-arrows to 3) • … • g₆₄ = 3 ↑^{g₆₃} 3 So Graham’s number is g₆₄. Even g₁ = 3 ↑↑↑↑ 3 is already so large that it dwarfs numbers like: • A googol (10¹⁰⁰) • A googolplex (10^googol) • Skewes’ number (another huge number from number theory) • The number of Planck volumes in the observable universe (~10⁸⁰ or so, depending on estimates) By g₂, you’re using g₁ (already insane) as the number of arrows.
By g₃, the number of arrows is itself defined by g₂, and so on, up to 64 levels. This is an example of extremely fast-growing recursion. The tower of operations is so deep that the number grows faster than any primitive recursive function — it enters the realm of functions that outpace the Ackermann function very quickly. Scale Comparisons (Still Inadequate) • Writing out Graham’s number in decimal digits would require more digits than there are atoms in the observable universe. • Even storing the number of digits of Graham’s number would require more space than the universe provides. • The last few digits of Graham’s number are computable (thanks to modular arithmetic tricks). For example, the last ten digits are …2464195387. But the vast majority of its digits are completely inaccessible. Historical Context & Legacy • For many years,
The American dream🇺🇸🦅 Four brilliant Actors give away hugs.🥰 #truecringecomunnity #fyp #viral #larp #tfd Graham’s number is one of the most extreme examples of a finite but unimaginably large number ever to appear in a legitimate mathematical proof. It was introduced by mathematician Ronald Graham in 1971 (joint work with Bruce Rothschild) and was popularized by Martin Gardner in his Scientific American column. The Mathematical Problem Behind It Graham’s number arises in Ramsey theory, which studies conditions under which order must appear in large enough structures, even if things look random. Specifically, the problem is about coloring the edges of a high-dimensional hypercube: • Take an n-dimensional hypercube (a generalization of a square in 2D or a cube in 3D). • Every pair of vertices (corners) is connected by an edge. • Color every edge either red or blue. The question: What is the smallest dimension n such that no matter how you color the edges, you are guaranteed to have a set of 4 coplanar vertices that form a complete graph (a “square” or K₄) where all edges are the same color (all red or all blue)? This is a Ramsey-type question: forcing monochromatic structures in large enough systems. • We know such an n exists (by Ramsey theory arguments). • The best known lower bound is relatively small (something like 13 or so — people have found colorings that avoid monochromatic planar K₄ up to certain dimensions). • Graham’s number was an upper bound: it proves that the monochromatic planar K₄ is guaranteed to appear by dimension Graham’s number (or much earlier). In other words, Graham’s number is a (very loose) upper bound for this Ramsey number. The actual value is almost certainly tiny compared to Graham’s number — probably under 100 or even much smaller — but proving tight bounds in Ramsey theory is notoriously hard. Why Is It So Big? The size comes from the recursive way it’s constructed using Knuth’s up-arrow notation, which extends ordinary arithmetic operations in a tower of increasingly powerful operations. Quick Refresher on Up-Arrow Notation • a ↑ b = a^b (exponentiation)
Example: 3 ↑ 3 = 27 • a ↑↑ b = a^(a^(a^…)) with b many a’s — a power tower of height b (tetration)
Example: 3 ↑↑ 3 = 3^(3^3) = 3^27 = 7,625,597,484,987 • a ↑↑↑ b = a ↑↑ (a ↑↑ (a ↑↑ …)) with b many a’s — a “tetration tower” evaluated from the top down or right-associatively. • a ↑↑↑↑ b = a ↑↑↑ (a ↑↑↑ … ) with b many a’s — and so on. Each additional arrow creates a massive jump in growth rate. Exact Definition of Graham’s Number Graham’s number is usually denoted as g₆₄ and is defined in 64 recursive steps: • Let g₁ = 3 ↑↑↑↑ 3
(3 with four up-arrows to 3) • g₂ = 3 ↑^{g₁} 3
(3 with g₁ many up-arrows to 3) • g₃ = 3 ↑^{g₂} 3
(3 with g₂ many up-arrows to 3) • … • g₆₄ = 3 ↑^{g₆₃} 3 So Graham’s number is g₆₄. Even g₁ = 3 ↑↑↑↑ 3 is already so large that it dwarfs numbers like: • A googol (10¹⁰⁰) • A googolplex (10^googol) • Skewes’ number (another huge number from number theory) • The number of Planck volumes in the observable universe (~10⁸⁰ or so, depending on estimates) By g₂, you’re using g₁ (already insane) as the number of arrows.
By g₃, the number of arrows is itself defined by g₂, and so on, up to 64 levels. This is an example of extremely fast-growing recursion. The tower of operations is so deep that the number grows faster than any primitive recursive function — it enters the realm of functions that outpace the Ackermann function very quickly. Scale Comparisons (Still Inadequate) • Writing out Graham’s number in decimal digits would require more digits than there are atoms in the observable universe. • Even storing the number of digits of Graham’s number would require more space than the universe provides. • The last few digits of Graham’s number are computable (thanks to modular arithmetic tricks). For example, the last ten digits are …2464195387. But the vast majority of its digits are completely inaccessible. Historical Context & Legacy • For many years,
introvert with terrible fomo
introvert with terrible fomo
Frag moments #punkpartner #FragPunkNewSeason #FragPunk #FragPunkPartnerShip
Frag moments #punkpartner #FragPunkNewSeason #FragPunk #FragPunkPartnerShip
— #sengen ] Took me 3700 years to finish #senku #asagirigen #drstone #anime
— #sengen ] Took me 3700 years to finish #senku #asagirigen #drstone #anime
早餐做这个鸡蛋堡，简单又好吃～ #美食 #美食分享 #food #Foodie #fyp
早餐做这个鸡蛋堡，简单又好吃～ #美食 #美食分享 #food #Foodie #fyp

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