--- title: "The Distribution of Primes and What Is Still Open" description: "Prove that the primes never run out, learn an estimate of how thinly they spread, and see what is still unknown." canonical: https://lightmysky.com/learn/mathematics/the-distribution-of-primes-and-what-is-still-open-mt_PJwm9kuQWQ source: https://lightmysky.com/learn/mathematics/the-distribution-of-primes-and-what-is-still-open-mt_PJwm9kuQWQ.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Distribution of Primes and What Is Still Open Prove that the primes never run out, learn an estimate of how thinly they spread, and see what is still unknown. Subject: Mathematics · Area: Number Theory · Ages 21 to 22 Page: https://lightmysky.com/learn/mathematics/the-distribution-of-primes-and-what-is-still-open-mt_PJwm9kuQWQ ## Ready when they can - Give Euclid's proof that the primes are infinite and one other proof of the same fact - State the prime number theorem and use it to estimate a prime count - Describe an open problem about primes precisely enough to say what a solution would need to establish ## Lesson: Primes never end but thin out A prime is an integer greater than 1 with exactly two positive divisors. Every N > 1 has a prime divisor: its least divisor greater than 1 cannot be composite. For a finite nonempty prime list, form its product P and N = P + 1. Each listed prime divides P and leaves remainder 1 in N, so none divides N. A prime divisor of N lies outside the list. Thus no finite list contains every prime. N need not be prime: 2 x 7 + 1 = 15 = 3 x 5. Write pi(x) for the number of primes at most x. For an exact count, exclude 1. Keep each sieving prime p and cross out its multiples starting at p squared. Every composite integer has a prime divisor at most its square root, so stop once p squared exceeds the bound. Up to 20, sieving with 2 and 3 leaves eight primes. To count primes in a < p <= b, subtract pi(a) from pi(b). For n >= 0, the Fermat number is F_n = 2^(2^n) + 1. These odd integers begin 3, 5, 17, 257. For n >= 1, F_0 x ... x F_(n-1) = F_n - 2. Prove this by induction, starting with 3 = 5 - 2. Multiplying the identity by F_n gives (A - 1)(A + 1) = A^2 - 1 = F_(n+1) - 2, where A = 2^(2^n). For m < n, any common positive divisor d of F_m and F_n divides the product F_0 x ... x F_(n-1) = F_n - 2. Since it also divides F_n, it divides 2. Both terms are odd, so d = 1. Thus distinct terms are coprime: they share no divisor greater than 1. Choose a prime divisor of each term. No prime can divide two terms, so infinitely many terms give infinitely many distinct primes. The Fermat numbers need not all be prime. For x > 1, ln x is the natural logarithm. The prime number theorem states that pi(x)/(x/ln x) tends to 1 as x tends to infinity. Thus x/ln x estimates the count of primes at most x, with relative error tending to zero. This limiting statement gives no numerical error bound at a chosen x. It neither gives an exact finite count nor lists the primes. **Example.** To estimate a count, enter x on a scientific calculator and use the ln function, not base-10 log. Divide x by that result and round only at the end. For x = 1000, ln x is about 6.90776, so x/ln x is about 144.765 and rounds to 145. The exact count is 168. This finite difference neither contradicts nor proves the limiting theorem. Divide the estimated count x/ln x by x to get 1/ln x. This estimates the fraction of integers from 1 to x that are prime. For example, ln x = 10 gives an estimated fraction of 1/10, or 10 percent. Multiply the fraction by 1000 to express it per thousand integers across the whole range. This does not predict the count in each short interval. Twin primes are two primes differing by 2, such as 5 and 7. The twin prime conjecture says there are infinitely many such pairs. It remains open. A proof would show that pairs occur beyond every bound. A disproof would show that only finitely many pairs exist. More examples, however large, do not settle the infinitude claim. The strong Goldbach conjecture says every even integer greater than 2 is a sum of two primes. The primes may be equal. It remains open. A proof must cover every even integer greater than 2. A disproof needs one such integer with no representation as a sum of two primes. Checking many examples does not prove this universal claim. **Recap.** Two proofs show infinitely many primes. The prime count is asymptotic to x/ln x. Twin-prime infinitude and strong Goldbach remain open. *Original lesson, practice questions and flashcards written for this dataset.* ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Divisibility and the Division Algorithm](https://lightmysky.com/learn/mathematics/divisibility-and-the-division-algorithm-mt_UkQNs-6-40) ## Optional connections - [Quadratic Residues and the Law of Reciprocity](https://lightmysky.com/learn/mathematics/quadratic-residues-and-the-law-of-reciprocity-mt_0KS9eTCY4p) - [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z)