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Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture ) or if there is a largest pair.
In this case, the Bateman–Horn conjecture reduces to the Hardy–Littlewood conjecture on the density of twin primes, according to which the number of twin prime pairs less than x is π 2 ( x ) ∼ 2 ∏ p ≥ 3 p ( p − 2 ) ( p − 1 ) 2 x ( log x ) 2 ≈ 1.32 x ( log x ) 2 . {\displaystyle \pi _{2}(x)\sim 2\prod _{p\geq 3}{\frac {p ...
The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let π 2 ( x ) {\displaystyle \pi _{2}(x)} denote the number of primes p ≤ x for which p + 2 is also prime (i.e. π 2 ( x ) {\displaystyle \pi _{2}(x)} is the number of twin primes with the smaller at most x ).
Goldbach's weak conjecture, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of Goldbach's conjecture. Ivan Vinogradov proved it for large enough n (Vinogradov's theorem) in 1937, [1] and Harald Helfgott extended this to a full proof of Goldbach's weak conjecture in 2013.
The set with an odd number of factors is just the primes between x 1/2 and x, so by the prime number theorem its size is (1 + o(1)) x / ln x. Thus these sieve methods are unable to give a useful upper bound for the first set, and overestimate the upper bound on the second set by a factor of 2.
A catadioptric optical system is one where refraction and reflection are combined in an optical system, usually via lenses and curved mirrors . Catadioptric combinations are used in focusing systems such as searchlights, headlamps, early lighthouse focusing systems, optical telescopes, microscopes, and telephoto lenses.
Two Leica oil immersion microscope objective lenses; left 100×, right 40×. The objective lens of a microscope is the one at the bottom near the sample. At its simplest, it is a very high-powered magnifying glass, with very short focal length. This is brought very close to the specimen being examined so that the light from the specimen comes ...
If the conjecture is true, then the gap size would be on the order of <. This also means there would be at least two primes between x 2 and (x + 1) 2 (one in the range from x 2 to x(x + 1) and the second in the range from x(x + 1) to (x + 1) 2), strengthening Legendre's conjecture that there is at least one prime in this range.