Mission: The Proof Detective — Level 8
Akuuu’s math teacher writes this on the board:
“Every even number greater than 2 can be written as the sum of two prime numbers.”
She gives examples: 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7 = 5+5
Then she says: “This has been checked for millions of numbers, but nobody has ever proven it always works.”
👉 How can something be true for millions of cases but still not proven? Is checking enough?
Think: If you check 1,000,000 cases and they all work, can you be 100% sure? What if case 1,000,001 breaks the rule?
✏️ Write your opinion: Is checking many examples the same as proving? Why or why not?
- 🔍 A conjecture is a guess based on patterns — it might be true, but isn’t proven
- ✅ A proof shows something is true for all cases, not just many
- ⚠️ Counter-example: One case that breaks a rule destroys the entire claim
Claim: “All numbers of the form n² − n + 41 are prime.”
Test: n=1 → 41 (prime) ✓
Test: n=2 → 43 (prime) ✓
👉 Test n=40 and n=41. Does the claim hold?
Akuuu’s friend claims: “The product of any three consecutive numbers is always divisible by 6.”
He tests: 1×2×3=6 ✓, 2×3×4=24 ✓, 3×4×5=60 ✓, 4×5×6=120 ✓
👉 Is testing 4 examples enough to trust this? What would a real proof need to show?
👉 Hint: Among any three consecutive numbers, what can you say about even numbers? About multiples of 3?
Claim: “n² + n + 41 is prime for all positive integers n.”
👉 Find the smallest n where this fails. Show your testing strategy.
A student “proves” that all odd numbers are prime:
“1 is prime (well, almost). 3 is prime. 5 is prime. 7 is prime. See? All odd numbers are prime!”
👉 What is wrong with this reasoning? Name at least two errors.
Claim: “The sum of any three consecutive integers is divisible by 3.”
👉 Prove this for ALL cases, not just examples. Use n, n+1, n+2.
🧠 Part A — Transfer
Before solving, consider this claim:
“For any whole number n, the number n² − 79n + 1601 is always prime.”
👉 This was tested for n = 0 to 79 and worked every time. Do you trust it? Predict: Is there a counter-example?
Now investigate:
👉 Find the counter-example. What does this famous example teach us about “tested many times”?
Think: This polynomial was specifically designed to generate primes for n=0 to 79. Yet it still fails eventually. What does this say about the difference between “tested” and “proven”?
🧠 Part B — Thinking Pause
- I questioned claims before accepting them
- I found counter-examples to destroy false claims
- I used algebra to prove something for all cases
⭐ Mission Reflection
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