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  2. Legendre's conjecture - Wikipedia

    en.wikipedia.org/wiki/Legendre's_conjecture

    Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between and (+) for every positive integer. [ 1 ] The conjecture is one of Landau's problems (1912) on prime numbers, and is one of many open problems on the spacing of prime numbers.

  3. Oppermann's conjecture - Wikipedia

    en.wikipedia.org/wiki/Oppermann's_conjecture

    and at least another prime between x 2 and x(x + 1). It can also be phrased equivalently as stating that the prime-counting function must take unequal values at the endpoints of each range. [3] That is: π (x 2 − x) < π (x 2) < π (x 2 + x) for x > 1. with π (x) being the number of prime numbers less than or equal to x.

  4. List of prime numbers - Wikipedia

    en.wikipedia.org/wiki/List_of_prime_numbers

    This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.

  5. Prime number theorem - Wikipedia

    en.wikipedia.org/wiki/Prime_number_theorem

    The prime number race generalizes to other moduli and is the subject of much research; Pál Turán asked whether it is always the case that π c,a (x) and π c,b (x) change places when a and b are coprime to c. [34]

  6. Formula for primes - Wikipedia

    en.wikipedia.org/wiki/Formula_for_primes

    However, it does not contain all the prime numbers, since the terms gcd(n + 1, a n) are always odd and so never equal to 2. 587 is the smallest prime (other than 2) not appearing in the first 10,000 outcomes that are different from 1. Nevertheless, in the same paper it was conjectured to contain all odd primes, even though it is rather inefficient.

  7. Arithmetic function - Wikipedia

    en.wikipedia.org/wiki/Arithmetic_function

    completely multiplicative if a(mn) = a(m)a(n) for all natural numbers m and n; Two whole numbers m and n are called coprime if their greatest common divisor is 1, that is, if there is no prime number that divides both of them. Then an arithmetic function a is additive if a(mn) = a(m) + a(n) for all coprime natural numbers m and n;

  8. Sieve of Atkin - Wikipedia

    en.wikipedia.org/wiki/Sieve_of_Atkin

    The following is pseudocode which combines Atkin's algorithms 3.1, 3.2, and 3.3 [1] by using a combined set s of all the numbers modulo 60 excluding those which are multiples of the prime numbers 2, 3, and 5, as per the algorithms, for a straightforward version of the algorithm that supports optional bit-packing of the wheel; although not specifically mentioned in the referenced paper, this ...

  9. Legendre symbol - Wikipedia

    en.wikipedia.org/wiki/Legendre_symbol

    In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1. Its value at zero is 0.