The Distribution of Primes and What Is Still Open · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Primes never end but thin out

Mathematics · Number Theory · ages 21-22
Name ______________________   Date ____________

  1. How many primes are at most 10?

    Answer: ______________

  2. Primes 2, 3, 5 are given. Euclid forms one plus their product. What number does Euclid form?

    • 30
    • 32
    • 15
    • 31

  3. Primes 2, 3, 5 are given. Euclid forms one plus their product. What number results?

    • 30
    • 32
    • 31

  4. Twin prime conjecture asks whether infinitely many prime pairs differ by 2, like 11 and 13. It remains unsolved.

    Circle one:   True   False

  5. Euclid shows any finite prime list misses a prime, since one plus the product has a prime divisor outside it.

    Circle one:   True   False

  6. Euclid proof shows any finite list of primes misses at least one prime because one plus their product has a prime divisor outside the list. Is this reasoning correct?

    Circle one:   True   False

  7. Prime number theorem estimates the count up to 100 as 100 divided by the natural log of 100. Rounded to the nearest whole number, what is this estimate?

    • 25
    • 22
    • 50
    • 10

  8. The theorem estimates the prime count up to 100 as 100 over natural log of 100. Rounded, what is it?

    • 25
    • 22
    • 50

  9. What would a solution to the twin prime conjecture have to establish?

    • A proof that some even number fails as a sum of two primes
    • A proof that twin pairs go on forever or stop somewhere
    • A faster sieve for small primes

  10. Prime number theorem estimates the count up to 1000 as 1000 divided by the natural log of 1000, about 144.8. Rounded to the nearest whole number, what is the estimate?

    Answer: ______________

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Answer key

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

Primes never end but thin out W1-mt_PJwm9kuQWQ-s1

  1. 4 · Primes up to 10 are 2, 3, 5, 7, a total of 4.
  2. 31 · 31 is correct since product 2 times 3 times 5 equals 30, plus one equals 31.
  3. 31 · Product 2 times 3 times 5 is 30, plus one is 31.
  4. True · Twin primes sit distance 2 apart, and infinitude is still open.
  5. True · That divisor cannot sit on the old list, since each old prime leaves remainder 1.
  6. True · Any prime divisor of the constructed number cannot be in the original list, so the list is incomplete.
  7. 22 · 22 is the rounded estimate since 100 divided by about 4.6 gives about 21.7, while 25 is the exact count and 50 and 10 are far off.
  8. 22 · Hundred over about 4.6 is about 21.7, rounding to 22, while 25 is the exact count.
  9. A proof that twin pairs go on forever or stop somewhere · Settling twins needs a verdict on infinitude, not examples or speed.
  10. 145 · 1000 divided by about 6.9 gives about 144.8, rounding to 145.
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