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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 10 n + d ( d = 1, 3, 7, 9) are primes ending in the decimal digit d .
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
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .
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]
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] Granville and Martin give a thorough exposition and survey. [33] Graph of the number of primes ending in 1, 3, 7, and 9 up to n ...
Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities). A prime number has Ω(n) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS). There are many special types of prime numbers. A composite number has Ω(n) > 1.
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d.
7 March 2021 4,447,272 67 4×3 9214845 + 1 10 September 2024 4,396,600 68 9145334×3 9145334 + 1 25 December 2023 4,363,441 69 4×5 6181673 – 1 15 July 2022 4,320,805 70 396101×2 14259638 – 1 3 February 2024 4,292,585 71 6962×31 2863120 – 1 29 February 2020 4,269,952 72 37×2 14166940 + 1 24 June 2022 4,264,676 73 99739×2 14019102 – 1