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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.
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 ).
The main topic of the book is the conjecture that there exist infinitely many twin primes, dating back at least to Alphonse de Polignac (who conjectured more generally in 1849 that every even number appears infinitely often as the difference between two primes), and the significant progress made recently by Yitang Zhang and others on this problem.
It is named after Paul Kirkpatrick and Albert Baez, the inventors of the X-ray microscope. [1] Although X-rays can be focused by compound refractive lenses, these also reduce the intensity of the beam and are therefore undesirable. KB mirrors, on the other hand, can focus beams to small spot sizes with minimal loss of intensity.
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 Chen primes are named after Chen Jingrun, who proved in 1966 that there are infinitely many such primes. This result would also follow from the truth of the twin prime conjecture as the lower member of a pair of twin primes is by definition a Chen prime. The first few Chen primes are
Lloyd's mirror is an optics experiment that was first described in 1834 by Humphrey Lloyd in the Transactions of the Royal Irish Academy. [1] Its original goal was to provide further evidence for the wave nature of light , beyond those provided by Thomas Young and Augustin-Jean Fresnel .
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.