به انگلیسی

Primes never end but thin out

Mathematics · Number Theory · ages 21-22

Name ______________________ Date ____________

For the prime list 2, 5, 11, the product is 110. What remainder does 110 + 1 leave when divided by the listed prime 5?
Answer: ______________
The prime list is 3, 5. Its product plus one is 16. Which prime divisor of 16 is missing from that list?
- 2
- 3
- 5
- 7
Primes 2, 3, 5 are given. Euclid forms one plus their product. What number results?
- 30
- 32
- 31
The twin prime conjecture asks whether infinitely many prime pairs differ by 2, like 11 and 13. It remains unsolved.
Circle one: True False
In Euclid's construction, a prime from the finite nonempty starting list can divide the product plus one.
Circle one: True False
Choosing a prime divisor from each term of an infinite sequence of pairwise coprime integers greater than 1 gives infinitely many distinct primes.
Circle one: True False
Let pi(x) count primes at most x. Which ratio tends to 1 as x tends to infinity, according to the prime number theorem?
- pi(x) / x
- pi(x) / (x / ln x)
- pi(x) / (x x ln x)
- pi(x) / (x / (ln x)^2)
The theorem estimates the prime count up to 100 as 100 divided by the natural log of 100. What is the rounded estimate?
- 25
- 22
- 50
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
For a large bound B, suppose ln B = 20. Using pi(B)/B approximately 1/ln B, express the estimated whole-range prime fraction per 1000 integers. What number results?
Answer: ______________

Answer key

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

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1 · 110 = 5 x 22, so 111 = 5 x 22 + 1. The remainder is 1, which shows that 5 does not divide 111.
2 · 16 = 2 x 8, so 2 is a prime divisor outside the list. A composite construction can still supply a new prime divisor.
31 · The product 2 times 3 times 5 is 30. Adding one gives 31.
True · Twin primes sit distance 2 apart, and infinitude is still open.
False · Each listed prime divides the product P. It leaves remainder 1 in P + 1, so none of the listed primes divides P + 1. A prime divisor of P + 1 must be outside the list.
True · Each term has a prime divisor. Pairwise coprimality prevents any prime from dividing two terms, so the chosen prime divisors are distinct.
pi(x) / (x / ln x) · The comparison function is x/ln x. The theorem says the prime count divided by that function tends to 1.
22 · 100 divided by about 4.6 is about 21.7, which rounds to 22. The exact count is 25.
A proof that twin pairs go on forever or stop somewhere · Settling the twin prime conjecture requires deciding whether there are infinitely many such pairs. Examples or a faster sieve do not settle it.
50 · The estimated fraction is 1/20. Multiplying by 1000 gives 50. This expresses a whole-range fraction per thousand, not a prediction for each block of 1000 integers.