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  • https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Barrus_and_Clark)/01%3A_Chapters/1.11%3A_Prime_Numbers
    This gives us a slight shortcut to finding primes with the Sieve of Eratosthenes: in our example above, once we have circled 7 and crossed out its multiples in the example above, every other number cu...This gives us a slight shortcut to finding primes with the Sieve of Eratosthenes: in our example above, once we have circled 7 and crossed out its multiples in the example above, every other number currently in the list that has not yet been circled or crossed out is guaranteed to be prime and can immediately be circled, since 7 is the largest prime number that is less than or equal to the square root of 100.
  • https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji)/02%3A_Prime_Numbers/2.07%3A_Theorems_and_Conjectures_involving_prime_numbers
    We have proved that there are infinitely many primes. We have also proved that there are arbitrary large gaps between primes. The question that arises naturally here is the following: Can we estimate ...We have proved that there are infinitely many primes. We have also proved that there are arbitrary large gaps between primes. The question that arises naturally here is the following: Can we estimate how many primes are there less than a given number? The theorem that answers this question is the prime number theorem.
  • https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Clark)/01%3A_Chapters/1.10%3A_Prime_Numbers
    An integer n2 is composite if and only if there are integers a and b such that n=ab, 1<a<n, and 1<b<n. If n is composite there is a positive integer a such that \(a...An integer n2 is composite if and only if there are integers a and b such that n=ab, 1<a<n, and 1<b<n. If n is composite there is a positive integer a such that a1, an and an. Prove that if a and n are positive integers such that n2 and an1 is prime then a must be 2.

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