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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.
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 ...
IMSL Numerical Libraries are libraries of numerical analysis functionality implemented in standard programming languages like C, Java, C# .NET, Fortran, and Python. The NAG Library is a collection of mathematical and statistical routines for multiple programming languages (C, C++, Fortran, Visual Basic, Java, Python and C#) and packages (MATLAB ...
In this case, the number of primitive Pythagorean triples (a, b, c) with a < b is 2 k−1, where k is the number of distinct prime factors of c. [25] There exist infinitely many Pythagorean triples with square numbers for both the hypotenuse c and the sum of the legs a + b.
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. [1] [2] It is denoted by π(x) (unrelated to the number π). A symmetric variant seen sometimes is π 0 (x), which is equal to π(x) − 1 ⁄ 2 if x is exactly a prime number, and equal to π(x) otherwise.
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.
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.
Consequently, a prime number divides at most one prime-exponent Mersenne number. [25] That is, the set of pernicious Mersenne numbers is pairwise coprime. If p and 2p + 1 are both prime (meaning that p is a Sophie Germain prime), and p is congruent to 3 (mod 4), then 2p + 1 divides 2 p − 1. [26]