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. Goldbach writes 28 as a sum of two primes. What is the smaller of the two?

    Answer: ______________

  2. A student lists Fermat's Last Theorem among open problems. Is that listing right?

    Circle one:   True   False

  3. Goldbach writes 28 as a sum of two primes. What is the smaller of the two?

    Answer: ______________

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

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

  5. A student says mathematicians have proved there are infinitely many twin primes. Is that right?

    Circle one:   True   False

  6. Chen's theorem is a famous partial result toward Goldbach. What does it say?

    • Every large even is the sum of two primes
    • Every large even is a prime plus a product of at most two primes
    • Every large even is the sum of three primes
    • Every large even is a perfect square plus one

  7. How many primes are there below 30?

    Answer: ______________

  8. A partial result settles Goldbach up to 10 to the 18. What remains?

    • Nothing: the conjecture is proved
    • All larger evens, with no general argument yet
    • A faster weekend sieve

  9. Sue states Goldbach for odd numbers as sums of two primes. What is wrong?

    • The even-sum version is the conjecture; the odd version is false
    • Products settle the question faster
    • The infinitude of primes already proves it

  10. A student states the Riemann hypothesis as: every nontrivial zero of zeta has real part 1/2. Is that precise statement right?

    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. 5 · Testing pairs upward gives 5 plus 23.
  2. False · Wiles proved it in the 1990s, so it now belongs to the solved list.
  3. 5 · Testing pairs from the smallest prime upward gives 5 + 23 = 28.
  4. Every even integer greater than 2 is the sum of two primes · The precise claim names even integers above 2 as prime sums.
  5. False · Infinitely many twin primes remains open; only bounded gaps are known.
  6. Every large even is a prime plus a product of at most two primes · Chen proved every large even is a prime plus a product of at most two primes, one step short of Goldbach.
  7. 10 · Sieving to 30 leaves 2, 3, 5, 7, 11, 13, 17, 19, 23, 29: ten primes.
  8. All larger evens, with no general argument yet · A bound plus a gap is exactly what partial means.
  9. The even-sum version is the conjecture; the odd version is false · Odds cannot split that way, so she stated a different, false claim.
  10. True · The hypothesis pins every nontrivial zero to the critical line with real part 1/2.
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