Fundamental Theorem of Arithmetic

Overview

A concise visual proof that every integer $n>1$ can be expressed as a product of prime numbers and that this prime factorisation is unique apart from the order of the factors. Viewers should come away understanding existence (via prime‑factor trees) and uniqueness (by comparing two different factorizations).

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Phases

#	Phase Name	Duration	Description
1	Intro	~4 s	Fade‑in the title “Fundamental Theorem of Arithmetic” and a brief animation of a number line highlighting the integer 60 as a representative example.
2	Statement	~5 s	Display the theorem statement in a caption band: “Every integer $n>1$ can be written uniquely as a product of primes (order ignored).” The caption fades in while the main area stays blank.
3	Existence (Prime‑Factor Tree)	~8 s	Build a factor tree for the highlighted number (e.g., 60 → 2·30 → 2·2·15 → 2·2·3·5). Each split is animated sequentially, with prime nodes highlighted in a distinct colour once reached.
4	Uniqueness (Two Competing Trees)	~8 s	Simultaneously grow two different factor trees for the same number (e.g., 60 = 2·2·3·5 and 60 = 2·3·2·5) using a split‑screen view within the main area. Animated arrows then slide the factors together, showing that after reordering the multisets of primes match exactly.
5	Summary	~4 s	Fade out the trees, leave only the prime multiset $\{2,2,3,5\}$ displayed, and circle‑highlight the caption “Unique up to order”. End with a brief pause before fade‑out.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                 MAIN (visual)               │
│                                             │
├─────────────────────────────────────────────┤
│ Caption / key statement (persistent footer) │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Number line, factor trees, prime‑multiset visual	Uses colour: primes = blue, composite splits = gray
Caption	The theorem statement and final “Unique up to order” label	Appears from Phase 2 onward; stays throughout

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Notes

• Assumptions: The audience is familiar with basic primes and multiplication; no explanatory text is needed beyond the visual steps.
• The example number (60) is chosen because it has multiple prime factors and allows a clear illustration of both existence and uniqueness.
• All fades and transforms are kept short (≈0.5 s) to stay within the ~30 s total duration.
• No additional on‑screen text is used except the caption band; all reasoning is conveyed visually.
• Colours are kept simple: blue for prime nodes, gray for intermediate composites, green for the final multiset highlight.