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Heegner number
What's so special about this number theorem?
Stark-Heegner theorem - Wikipedia, the free encyclopedia
It is apparent that the connection between prime numbers and imaginary numbers is rather interesting.
Many know that:
* Plasma is fractal in nature. Plasma cosmologists often refer to a cosmic "triple jump".
* The two most famous fractals, the Mandelbrot set and the Julia set are related by complex numbers.
* The complex plane, as well as quaternions, as well as octonions, have connections to so called "imaginary" numbers.
* "Imaginary" numbers are also used in the Steinmetz method of electrical engineering calculation, often referenced by Eric Dollard.
* A field of complex eigenvectors [α,β] where λ = α ± i*β can encode both sink/source (divergence and compressibility) via α as well as spin (curl and vorticity) via β. One can study eigenvectors as it pertains to linear algebra. A "non-linear system" is indistinguishable from a sufficiently fractal linear system. Thus a fractal linear system of equations can exist, with seeming "non-linearities" being the result of the hard to predict, hard to formulate, yet deterministic nature of fractal chaos, which consists of an infinity of terms. See discussion at Is non-linearity incontrovertible? What about hidden variables?.
Speaking of quaternions and octonions, mainstream mathematicians have discovered a kind of "magic square" (in a loose sense of the term) which has significance in number theory. It's (uncannily) called the "Freudenthal–Tits magic square".
Freudenthal magic square - Wikipedia, the free encyclopedia
Now, the word "Lie" is pronounced "Lee". It refers to someone's last name.
See for example, similar names:
They are all pronounced in the same way, as far as the level precision of American accents are concerned.
(Note the use of letters L, e, & i. L, e, i, E, & I are all important symbols used in Eric P. Dollard's laws of electro-magnetic induction. Even more uncannily, epsilon_0*E + P = D, where E is the electric field, P is the polarization field, and D is the electric displacement field. Eric Dollard also accepts BitCoin.)
Maybe this is the real magic square we should be concerned about. Now, I do believe in Searl's story, but I think there is more. I think it is possible that what Searl built was a receiver which may work provided that an external progenitor provides the energy source. This is also how I view extraterrestrial phenomenon taking place in Area 51. Any advanced Area 51 technology capable of "ufo" type maneuvers is in my opinion actually the result of the devices' ability to receive energy artificially transmitted to them by extraterrestrials. The ETs can simply remove the source of the power when they decide to pull the plug on such black projects. In other words, I believe that shady characters at Area 51 are being handed a kind of Trojan Horse as well as fake "inter-dimensional physics", for the sake of the rest of mankind (in opposition to those in financial control of the black projects), so that at the right time and place, the secrets of UFOs can be revealed, with the "99%" with the technological advantage, resulting in Area 51 with a bunch of essentially useless pseudo-science. Such is the fate of people who failed the same type of tests that Jesus and Job (of the Bible) passed, tests which were given to them by a proctor, or "Devil's advocate", specifically Satan.
Not much in our universe can be reduced to coincidence. Consider the following:
Talk:Lucky numbers of Euler - Wikipedia, the free encyclopedia
Another interesting fact?
Take the letters in FREUDENTHAL:
6+18+5+21+4+5+14+20+8+1+12 = 114
MAGIC:
13+1+7+9+3 = 33
SQUARE:
19+17+21+1+18+5 = 81 = 9^2
MAGIC SQUARE:
13+1+7+9+3 + 19+17+21+1+18+5 = 33 + 81 = 114
If you find yourself seeing similar strange "coincidences" on a regular basis, then you are certainly not alone.
In number theory, a Heegner number is a square-free positive integer d such that the imaginary quadratic field Q(√−d) has class number 1. Equivalently, its ring of integers has unique factorization.[1]
The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory.
According to the Stark–Heegner theorem there are precisely nine Heegner numbers:
1, 2, 3, 7, 11, 19, 43, 67, 163.
This result was conjectured by Gauss and proven by Kurt Heegner in 1952.
Euler's prime-generating polynomial
n^2 - n + 41, \, ,
which gives (distinct) primes for n = 1, ..., 40, is related to the Heegner number 163 = 4 · 41 − 1.
Euler's formula, with n taking the values 1,... 40 is equivalent to
n^2 + n + 41, \,
with n taking the values 0,... 39, and Rabinowitz[2] proved that
n^2 + n + p \,
gives primes for n=0,\dots,p-2 if and only if its discriminant 1-4p equals minus a Heegner number.
(Note that p-1 yields p^2, so p-2 is maximal.) 1, 2, and 3 are not of the required form, so the Heegner numbers that work are 7, 11, 19, 43, 67, 163, yielding prime generating functions of Euler's form for 2,3,5,11,17,41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.[3]
The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory.
According to the Stark–Heegner theorem there are precisely nine Heegner numbers:
1, 2, 3, 7, 11, 19, 43, 67, 163.
This result was conjectured by Gauss and proven by Kurt Heegner in 1952.
Euler's prime-generating polynomial
n^2 - n + 41, \, ,
which gives (distinct) primes for n = 1, ..., 40, is related to the Heegner number 163 = 4 · 41 − 1.
Euler's formula, with n taking the values 1,... 40 is equivalent to
n^2 + n + 41, \,
with n taking the values 0,... 39, and Rabinowitz[2] proved that
n^2 + n + p \,
gives primes for n=0,\dots,p-2 if and only if its discriminant 1-4p equals minus a Heegner number.
(Note that p-1 yields p^2, so p-2 is maximal.) 1, 2, and 3 are not of the required form, so the Heegner numbers that work are 7, 11, 19, 43, 67, 163, yielding prime generating functions of Euler's form for 2,3,5,11,17,41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.[3]
Stark-Heegner theorem - Wikipedia, the free encyclopedia
In number theory, a branch of mathematics, the Stark–Heegner theorem states precisely which quadratic imaginary number fields admit unique factorisation in their ring of integers. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number.
Let Q denote the set of rational numbers, and let d be a square-free integer (i.e., a product of distinct primes) other than 1. Then Q(√d) is a finite extension of Q, called a quadratic extension. The class number of Q(√d) is the number of equivalence classes of ideals of the ring of integers of Q(√d), where two ideals I and J are equivalent if and only if there exist principal ideals (a) and (b) such that (a)I = (b)J. Thus, the ring of integers of Q(√d) is a principal ideal domain (and hence a unique factorization domain) if and only if the class number of Q(√d) is equal to 1. The Stark–Heegner theorem can then be stated as follows:
If d < 0, then the class number of Q(√d) is equal to 1 if and only if
d \in \{\, -1, -2, -3, -7, -11, -19, -43, -67, -163\,\}.
These are known as the Heegner numbers.
Let Q denote the set of rational numbers, and let d be a square-free integer (i.e., a product of distinct primes) other than 1. Then Q(√d) is a finite extension of Q, called a quadratic extension. The class number of Q(√d) is the number of equivalence classes of ideals of the ring of integers of Q(√d), where two ideals I and J are equivalent if and only if there exist principal ideals (a) and (b) such that (a)I = (b)J. Thus, the ring of integers of Q(√d) is a principal ideal domain (and hence a unique factorization domain) if and only if the class number of Q(√d) is equal to 1. The Stark–Heegner theorem can then be stated as follows:
If d < 0, then the class number of Q(√d) is equal to 1 if and only if
d \in \{\, -1, -2, -3, -7, -11, -19, -43, -67, -163\,\}.
These are known as the Heegner numbers.
Many know that:
* Plasma is fractal in nature. Plasma cosmologists often refer to a cosmic "triple jump".
* The two most famous fractals, the Mandelbrot set and the Julia set are related by complex numbers.
* The complex plane, as well as quaternions, as well as octonions, have connections to so called "imaginary" numbers.
* "Imaginary" numbers are also used in the Steinmetz method of electrical engineering calculation, often referenced by Eric Dollard.
* A field of complex eigenvectors [α,β] where λ = α ± i*β can encode both sink/source (divergence and compressibility) via α as well as spin (curl and vorticity) via β. One can study eigenvectors as it pertains to linear algebra. A "non-linear system" is indistinguishable from a sufficiently fractal linear system. Thus a fractal linear system of equations can exist, with seeming "non-linearities" being the result of the hard to predict, hard to formulate, yet deterministic nature of fractal chaos, which consists of an infinity of terms. See discussion at Is non-linearity incontrovertible? What about hidden variables?.
Speaking of quaternions and octonions, mainstream mathematicians have discovered a kind of "magic square" (in a loose sense of the term) which has significance in number theory. It's (uncannily) called the "Freudenthal–Tits magic square".
Freudenthal magic square - Wikipedia, the free encyclopedia
In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie algebra to a pair of division algebras A, B. The resulting Lie algebras have Dynkin diagrams according to the table at right. The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in A and B, despite the original construction not being symmetric, though Vinberg's symmetric method gives a symmetric construction; it is not a magic square as in recreational mathematics.
The Freudenthal magic square includes all of the exceptional Lie groups apart from G2, and it provides one possible approach to justify the assertion that "the exceptional Lie groups all exist because of the octonions": G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector space – see prehomogeneous vector space).
The Freudenthal magic square includes all of the exceptional Lie groups apart from G2, and it provides one possible approach to justify the assertion that "the exceptional Lie groups all exist because of the octonions": G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector space – see prehomogeneous vector space).
See for example, similar names:
They are all pronounced in the same way, as far as the level precision of American accents are concerned.
(Note the use of letters L, e, & i. L, e, i, E, & I are all important symbols used in Eric P. Dollard's laws of electro-magnetic induction. Even more uncannily, epsilon_0*E + P = D, where E is the electric field, P is the polarization field, and D is the electric displacement field. Eric Dollard also accepts BitCoin.)
Maybe this is the real magic square we should be concerned about. Now, I do believe in Searl's story, but I think there is more. I think it is possible that what Searl built was a receiver which may work provided that an external progenitor provides the energy source. This is also how I view extraterrestrial phenomenon taking place in Area 51. Any advanced Area 51 technology capable of "ufo" type maneuvers is in my opinion actually the result of the devices' ability to receive energy artificially transmitted to them by extraterrestrials. The ETs can simply remove the source of the power when they decide to pull the plug on such black projects. In other words, I believe that shady characters at Area 51 are being handed a kind of Trojan Horse as well as fake "inter-dimensional physics", for the sake of the rest of mankind (in opposition to those in financial control of the black projects), so that at the right time and place, the secrets of UFOs can be revealed, with the "99%" with the technological advantage, resulting in Area 51 with a bunch of essentially useless pseudo-science. Such is the fate of people who failed the same type of tests that Jesus and Job (of the Bible) passed, tests which were given to them by a proctor, or "Devil's advocate", specifically Satan.
Not much in our universe can be reduced to coincidence. Consider the following:
Talk:Lucky numbers of Euler - Wikipedia, the free encyclopedia
Originally posted by kmarinas86
Take the letters in FREUDENTHAL:
6+18+5+21+4+5+14+20+8+1+12 = 114
MAGIC:
13+1+7+9+3 = 33
SQUARE:
19+17+21+1+18+5 = 81 = 9^2
MAGIC SQUARE:
13+1+7+9+3 + 19+17+21+1+18+5 = 33 + 81 = 114
If you find yourself seeing similar strange "coincidences" on a regular basis, then you are certainly not alone.