Practice question · Multiple choice
Checking that a claim about all integers holds for the first ten thousand cases is not a proof, while a single failing case is a complete refutation. Why is the evidence so lopsided?
Hints
- How many cases does "for all n" cover, and how many have you checked?
- How many cases does it take to make "for all n" false?
Show the answer
D. Because verified cases leave infinitely many untested, and one falsifies.
Why
'For all n' is a conjunction of infinitely many claims, so verifying finitely many leaves infinitely many untested however large the finite number; its negation needs one witness. The history is genuinely alarming: Pólya's conjecture holds below roughly 900 million and fails after, which is why no finite threshold could ever settle it.
Practise Direct Proof and Counterexample
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More questions on Direct Proof and Counterexample
- A claim that holds for the first one million integers is thereby proven for all integers.
- Order the steps of a direct proof that the sum of two even integers is even.
- Claim: 'every prime number is odd'. Select every value that is a genuine COUNTEREXAMPLE.
- Sort each claim by what it would take to settle it.
- Every even number above 2 checked so far is the sum of two primes, verified past 4 × 10¹⁸. Why is Goldbach's…