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Rieman zeta funtion u + ív = zeta(x + íy) is a one-variable complex-vlaued holomorphic function defined at
С - { 1 }
define as a functional equation by:
u(x, y) + í v(x, y) = ζ(x, y) = ζ(x + í y)
ζ(x + íy) = 2(x + iy) * π(x + íy - 1) sin((x + í y) * π/ 2) Γ(1 - x - íy) ζ(1 - x - íy) when x + í y ∈ С - { 1 }
This is,
ζ(x + í y) / ζ(1 - x - íy) = 2(x + iy) * π(x + íy - 1) sin((x + í y) * π/ 2) Γ(1 - x - íy) ζ(1 - x - íy) when x + í y ≠ 1
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2, 3, 5, 7,
11, 13, 17, 19,
23, 29, 31, 37,
and more
A pair {a, a + 2} of two prime number is said to be a twin prime pair.
For examples, the pairs { 3, 5}, {5 7}, {11, 13}, {17, 19}, {29, 31}, {41, 43, {59, 61}, {71, 73} are twin prime pairs.
2 is the unique even prime number.If n > 2 and n is an even integer, then n is not prime.
And so 4, 6, 8, 10, 12, 14, 16, 18, 20, ... , 2k, ... are not prime numbers.
If p and q are prime numbers, p < q, and the is no prime number between them, their difference q - p is calld their prime gap.
3 - 2 =1, 5- 3 = 2, 7 - 5 = 2, 11- 7 = 4;, 13 - 11 = 2, 19 - 13 = 4, 19 - 17 = 2, 23 - 17 = 4, .....
And so 1, 2, 2, 4, 2, 4, 2, 4, 4, ... are prime gaps.
1, 2, 4, 6, 8, 14, 18, 20, are maximal prime gaps.
1 = 3 - 2,
2 = 5 - 3,
4 = 11 - 7
6 = 29 - 23.
8 = 97 - 89,
14 = 127 - 113,
18 = 541 - 523,
20 = 907 - 887,
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