Search results
Results From The WOW.Com Content Network
The primes of the form 2n+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4n+1 are Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes (with 2 omitted). The classes 10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d.
A number where some but not all prime factors have multiplicity above 1 is neither square-free nor squareful. The Liouville function λ(n) is 1 if Ω(n) is even, and is -1 if Ω(n) is odd. The Möbius function μ(n) is 0 if n is not square-free. Otherwise μ(n) is 1 if Ω(n) is even, and is −1 if Ω(n) is odd.
Therefore, every prime number other than 2 is an odd number, and is called an odd prime. [10] Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9.
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. [1] [2] The exponents p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 2 11 − 1 = 2047 = 23 × 89. [3]
By contraposition, if 2 p − 1 is prime then p is prime. If p is an odd prime, then every prime q that divides 2 p − 1 must be 1 plus a multiple of 2p. This holds even when 2 p − 1 is prime. For example, 2 5 − 1 = 31 is prime, and 31 = 1 + 3 × (2 × 5). A composite example is 2 11 − 1 = 23 × 89, where 23 = 1 + (2 × 11) and 89 = 1 ...
A prime number p of the form = + where q is an odd prime. A000979: Wieferich primes: 1093, 3511 Primes satisfying 2 p−1 ≡ 1 (mod p 2). A001220: Sophie Germain primes: 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, ... A prime number p such that 2p + 1 is also prime. A005384: Wilson primes: 5, 13, 563
However, the number of Pythagorean primes up to is frequently somewhat smaller than the number of non-Pythagorean primes; this phenomenon is known as Chebyshev's bias. [1] For example, the only values of n {\displaystyle n} up to 600000 for which there are more Pythagorean than non-Pythagorean odd primes less than or equal to n are 26861 and 26862.
The sieve starts with a list of the integers from 1 to n. From this list, all numbers of the form i + j + 2ij are removed, where i and j are positive integers such that 1 ≤ i ≤ j and i + j + 2ij ≤ n. The remaining numbers are doubled and incremented by one, giving a list of the odd prime numbers (that is, all primes except 2) below 2n + 2.