Open Problems as a Genre · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

What makes a question open

Mathematics · Mathematical Thinking · ages 23-24
Name ______________________   Date ____________

  1. For the Goldbach pair 28 = 11 + p, what is p?

    Answer: ______________

  2. Goldbach permits equal primes, so 6 = 3 + 3 is a valid instance of its claim.

    Circle one:   True   False

  3. How many unordered prime pairs sum to 18? Count p + q and q + p once. Repeated primes are allowed.

    Answer: ______________

  4. Which statement is Goldbach's conjecture, stated precisely?

    • Every odd integer greater than 2 is the sum of two primes
    • Every even integer greater than 2 is the product of two primes
    • Every even integer greater than 2 is the sum of two primes

  5. A theorem gives infinitely many prime pairs with gaps at most 1,000. By itself, this proves infinitely many twin primes.

    Circle one:   True   False

  6. Why does Chen's theorem leave a gap before Goldbach's full claim?

    • It supplies two primes for every even integer but leaves their order in each pair unspecified
    • It permits a composite second term and covers only sufficiently large evens
    • It applies to every odd integer instead of even integers and requires the same two-prime conclusion
    • It counts primes below each input instead of proving an additive representation for any input

  7. The identity 60 = 11 + 49 has the form allowed by Chen's theorem. How many prime factors does 49 have, counting repeats?

    Answer: ______________

  8. Suppose a verified computation checks Goldbach for every even integer from 4 through 10^18. What lies outside this result?

    • No even integers remain outside it, because its finite upper bound is so large
    • All even integers above 10^18 remain outside this computation
    • Only odd integers remain outside it, because every larger integer is odd

  9. Sue claims every odd integer greater than 2 is a sum of two primes. Which check refutes her claim?

    • 11: its required partner to 2 is 9, which is composite
    • 9: being composite prevents it from being a sum of any two prime numbers
    • 7: its pair 2 + 5 is disallowed because one of those two primes is even

  10. A claim is proved independent of specified axioms A. This establishes that no stronger axiom system can decide the claim.

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

What makes a question open W1-mt_qBWcaUWG9J-s1

  1. 17 · Subtract the given term: p = 28 - 11 = 17. Both 11 and 17 are prime.
  2. True · The integer 6 is even and greater than 2. Both terms are prime, and Goldbach does not require different primes.
  3. 2 · The complete search gives two pairs: 5 + 13 and 7 + 11.
  4. Every even integer greater than 2 is the sum of two primes · Goldbach concerns even integers greater than 2 and asks for a sum of two primes.
  5. False · Twin primes require a gap of exactly 2. A bound of 1,000 permits other gaps and does not by itself establish infinitely many gap-2 pairs.
  6. It permits a composite second term and covers only sufficiently large evens · Chen allows a product of two primes and covers sufficiently large evens. Goldbach requires two primes for every even integer greater than 2.
  7. 2 · 49 = 7 × 7 contains two prime factors counting repetition. This illustrates the allowed form, not a Goldbach pair or Chen's threshold.
  8. All even integers above 10^18 remain outside this computation · The computation stops at 10^18. Even integers above that bound are outside its stated coverage.
  9. 11: its required partner to 2 is 9, which is composite · An odd prime sum must use 2. For 11, the other term would be 9, which is composite. In contrast, 9 = 2 + 7 and 7 = 2 + 5.
  10. False · Independence from A concerns proofs from A. It does not establish that every stronger axiom system has the same limitation.
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