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Prime gap probability density for primes up to 1 million. Peaks occur at multiples of 6. [1]A prime gap is the difference between two successive prime numbers.The n-th prime gap, denoted g n or g(p n) is the difference between the (n + 1)-st and the n-th prime numbers, i.e.
All prime numbers from 31 to 6,469,693,189 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences". MathWorld.
Andrica's conjecture (named after Romanian mathematician Dorin Andrica ) is a conjecture regarding the gaps between prime numbers. [1] The conjecture states that the inequality + < holds for all , where is the nth prime number.
By using a table of maximal gaps, Farideh Firoozbakht verified her conjecture up to 4.444 × 10 12. [2] Now with more extensive tables of maximal gaps, the conjecture has been verified for all primes below 2 64 ≈ 1.84 × 10 19. [3] [4] [5] If the conjecture were true, then the prime gap function = + would satisfy: [6]
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, [1] is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be.
A counterexample near that size would require a prime gap a hundred million times the size of the average gap. Järviniemi, [ 22 ] improving on work by Heath-Brown [ 23 ] and by Matomäki, [ 24 ] shows that there are at most x 7 / 100 + ε {\displaystyle x^{7/100+\varepsilon }} exceptional primes followed by gaps larger than 2 p {\displaystyle ...
Others include Bertrand's postulate, on the existence of a prime between and , Oppermann's conjecture on the existence of primes between , (+), and (+), Andrica's conjecture and Brocard's conjecture on the existence of primes between squares of consecutive primes, and Cramér's conjecture that the gaps are always much smaller, of the order ().
For n = 2, it is the twin prime conjecture. For n = 4, it says there are infinitely many cousin primes (p, p + 4). For n = 6, it says there are infinitely many sexy primes (p, p + 6) with no prime between p and p + 6. Dickson's conjecture generalizes Polignac's conjecture to cover all prime constellations.