[https://github.com/gabrielc42/q-and1-p-characters-of-Prime-numbers/blob/main/stuff.md
How Fish see the world...
I started playing with numbers and got deeply fascinated with what I was finding. Have yet to read through thoroughly enough of the many sources, but they are not haphazardly inputted into this. And of course, some of this may seem contrived and incorrect. However, our lives are too short to restrict ourselves the abundance of the universe and dwaddle with trivial mistakes, especially when it comes to enjoying math. I hope to continue research with these topics, of all the key words listed below. Also please note that I do not have an undergrad degree in math nor physics (yet). Love math and physics!
I also recently learned about these and wonder about the at least correlation.
# Characteristics of numbers 11, 7, 3, 5675055181, 13, 1753, 56750551851, 17, 29, 19, 2, 5, 4273, 127, 23 and other transcendental numbers under Bernoulli distribution of influence of sets and Likelihood function, to discuss physical laws of prime numbers, imaginary partition functions, and the Riemann hypothesis. Continued reflection on proof by Laurent series and Likelihood function. Latter conjecture about Electroweak fields, fluid dynamics, and holomorphism.
In this discussion I present a series of equatins, arithmetic, and various utilities in order to studying a multitude of systems. I begin with pertitent information regarding origin points, but in reality I began with just numbers. Continuous reasoning and findings of prime numbers alongside partition review led me to discover for a another time the significance of prime numbers and partitions, including the zeta function. I brought about several research topics that deal with physical laws and applications between these numbers and characteristics, as well as methodologies simply derived and interpreted by myself and hopefully following some rational sound math formula and strategies in order to rediscover through a similar path an array of transcendental numbers, including a lost one, dealing with other various intricacies and multiplots of prime numbers, special formulas and conjectures, all in order to study systems that exhibit quantum mechanical behavior, such as some particles. Imaginary thermodynamics anyone?
In this paper , https://link.springer.com/chapter/10.1007/978-3-642-87140-5_2 the proof of the important Orlicz- Pettis theorem which claims the equivalence of weak and strong unconditional convergence of series in Banach spaces. The second paragraph contains Riemann’s theorem which asserts that absolute and unconditional convergence of series in finite dimensional vector spaces are the same, and the famous Dvoretzky-Rogers theorem. The latter states the existence in every infinite dimensional Banach space of an unconditional series which is not absolutely convergent, a fact, which has been conjectured for about twenty years and which has been settled down by Dvoretzky and Rogers in 1950.
In this PHD dissertation, Convergence of Frame Series from Hilbert spaces to Banach spaces And l^1-boundedness, https://repository.gatech.edu/entities/publication/36f42a49-2e32-4cb4-9f3e-a6196c623a47, similar methods and information are mentioned.
Calculation of Gibbs partition function with imaginary time evolution on near-term quantum computers https://arxiv.org/pdf/2109.14880
<> 59.99 value calculated </>
<>
<> == 99999750062433397480084412018728, not a prime </>
<>
<> == 3659261913760489 -> is divisible by 23, a prime number </>
<> == 3829365617345466730201708252119 is not a prime </>
<> base: 59.999 — n=/=0 and this constant... This constant 8.0365 ,
1 / (1 - a(p)/(p))
1 / (1 - 23(8.0365)/(8.0365)) = -0.04545454545454545454545454545455
Graph this, {.567055181, .5052250171} graph in wolfram alpha, </>
- 11 ( - 7 - 3, - 5675055181 - 13 - 1753 - 56750551851 - 17 - 29 (1) - 19 (1) - 2 (1) - 5 (1) - 4273 - 127 - 23 (1) 14 dimensions, prime, their dreams<>
<>
<> The entropy is maximized when p = 0.5, indicating the highest level of uncertainty when both outcomes are equally likely. The entropy is zero when or p=0, or p = 1., where one outcome is certain. </>
https://mathworld.wolfram.com/PrimePartition.html
`` *key words: radio waves, partitions, prime numbers characteristics, Riemann hypothesis, Bernoulli distribution, Likelihood function, Euler product, meromorphic function, sets, summation, Laurent series, addition, series, pattern, gravity, imaginary numbers, complex analysis, isolated singularities, entropy, Agoh–Giuga conjecture, Banach space, Prime partition, Guiga number, Carmichael number, number theory, topological quantum matter, manifolds, Fourier transformation, Laplace transformation, complex differentiables, Gauss equation, Ramanujan, Euler, imaginary partition function, transcendental numbers * ``
970,462.32026163842254755194171738 95426903.18473885
In the 19th century, physicists developed the methods of statistical mechanics for studying many-particle systems, whereas mathematicians proved the distribution law for prime numbers. It turns out that the two apparently different approaches can be traced back to the same mathematical root, namely, the notion of partition function. In modern quantum field theory, the Feynman functional integral can be viewed as a partition function, as we will discuss later on. The typical procedure proceeds in the following two steps.
The imaginary partition function is useful in studying systems that exhibit quantum mechanical behavior, such as atoms and molecules
<h4> </h4>
TY - BOOK
AU - Zeidler, Eberhard
PY - 2006/01/01
SP -
N2 - This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists ranging from advanced undergraduate students to professional scientists. The book tries to bridge the existing gap between the different languages used by mathematicians and physicists. For students of mathematics it is shown that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which is beyond the usual curriculum in physics. It is the author's goal to present the state of the art of realizing Einstein's dream of a unified theory for the four fundamental forces in the universe (gravitational, electromagnetic, strong, and weak interaction). From the reviews: ". Quantum field theory is one of the great intellectual edifices in the history of human thought. This volume differs from other books on quantum field theory in its greater emphasis on the interaction of physics with mathematics. an impressive work of scholarship." (William G. Faris, SIAM Review, Vol. 50 (2), 2008) ". it is a fun book for practicing quantum field theorists to browse, and it may be similarly enjoyed by mathematical colleagues. Its ultimate value may lie in encouraging students to enter this challenging interdisciplinary area of mathematics and physics. Summing Up: Recommended. Upper-division undergraduates through faculty.".
SN - 978-3-540-34762-0
T1 - Quantum Field Theory I: Basics in Mathematics and Physics
VL -
DO - 10.1007/978-3-540-34764-4
JO - Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists
ER -
It's my lucky day. I solved for prime numbers. I was messing around with equations and papers and homework and math. Got into prime numbers based off ideas of 10 digit numbers and divisibility, and into Bernoulli’s distribution. I went through [partitions]( https://scholar.google.com/citations?view_op=view_citation&hl=en&user=4wpjDroAAAAJ&citation_for_view=4wpjDroAAAAJ:u5HHmVD_uO8C), into [Bernoullis distribution](https://en.wikipedia.org/wiki/Bernoulli_distribution), and [numbers](https://en.wikipedia.org/wiki/Bernoulli_number), from [Fermat’s little theorem](https://en.wikipedia.org/wiki/Proofs_of_Fermat's_little_theorem), and unto of course my flavor into the interpretations and sudo math, including...
q = 1 - p since p − 1 ≡ −1 (mod p) q = -1
-1 not trivial zero equality between a sum and a product 1 sum = product q = -1 q = -1 * n : inf 1 - p = -1 * n (1-p)/n = -1 -1/n - p/n = -11/n + p/n = 1
p - 1 = -1 mod p p = 0
1/n + 0/n = 1 n =/= 0
1/n + 0/n - 1 = 0 1((x/n) + 0/n - (x)) = 0
1 ((xn^[1/2]) + 0/n - (x)) = 0 n = {Integral{log(e)}}
the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2
x = sqrt(x^2 + 1)
x = {0, |x| - 1}
x = {0, (|log(e)| - 1) * } = {0, .5657055181}
{.567055181, .5052250171}
{b,c} graph
b is divisible by 7, c is divisible by 11
11 and 7 are prime numbers
11 and 7 at integer magnitude = b and c
b / 7 = 81,007,883
refract at magnitude change: .81007883
c / 11 = 205,004,561 : 2.05004561
remember |log(e)| - 1 = .5675055181 .56570055 = 56570055 is divisible by 3 .5675955181 = 5675955181 is divisible by 13 .567055181 = 567055181 is divisible by 7 x = log(e) e * pi = 8.5397 .81007883 + .5675055181 = 8.5397 - .5675055181 = 7.972287 , 8.1007883 == 8.0365 ——————————
This constant 8.0365 will be valuable later on. It is a base off our n=/=0 earlier. Providing at least one value is all we need in order to understand what the root of the partition, n, is. This will explain the patterns, physical applications, and unique prime characteristics that follow the 19th century in which physicists developed the methods of statistical mechanics for studying many-particle systems, whereas mathematicians proved the distribution law for prime numbers. It turns out that the two apparently different approaches can be traced back to the same mathematical root, namely, the notion of partition function. In modern quantum field theory, the Feynman functional integral can be viewed as a partition function, as we will discuss later on.
https://en.wikipedia.org/wiki/Partition_of_a_set
Characteristics of prime numbers 11, 7, 3, 5675055181, 13, 1753, 56750551851, 17, 29, 2, 5, 4273, 127
https://socratic.org/questions/587d0569b72cff585cd949
integral(sqrt(x^2 + 1)dx) = (ln|x+(sqrt(x^2 + 1))| + x(sqrt(x^2 + 1)))/2 + c = sqrt(x^2 + 1)
x = sqrt(x^2 + 1) x =11, 7, … any other prime numbers? Is what is below similar to what is above? It needs to be proved.
11 >> 11.04536101718726 == 1104536101718726 , is divisible by 2 a prime number
7 >> 7.071067811865475 == 7071067811865475 is divisible by 5 a prime number
Occams razor, probabilistic determination here.
567055181 is divisible by 7, a prime number 81007883 is divisible by 11, a prime number 7972287, 56570055, 56750551851, 5,675,055,243 is divisible by 3, a prime number 5675955181 is divisible by 13, a prime number 1442695040888963 is divisible by 17, a prime number 7071067811865475 is divisible by 5, a prime number
56750551851 is prime 29 is prime
7,364,353 is divisible by 1753, a prime number
<h4>what are three solutions to these:</h4>
let’s coordinate based on the low prime numbers:
and remember, based on the equations above, there still remains more prime numbers to solve
for by arithmetic
a1 :: 7 =
!
b1 :: 11
c1 :: 3
d1 :: 13
it isnt in order, and our scale is constantly growing. continue on with this subseries, a box if u will. apply phyiscal laws and let’s also dicsuss the perturbation of electromagnetic field via flame ore fire:
-
ii = 0.207879576... (Here i is the imaginary number sqrt(-1). Isn't this a real beauty? How many people have actually considered rasing i to the i power? If a is algebraic and b is algebraic but irrational then ab is transcendental. Since i is algebraic but irrational, the theorem applies.
i^i is equal to e(- pi / 2 ) and several other values. Consider i^i = e(i log i ) = e( i times i pi / 2 ) . Since log is multivalued, there are other possible values for i^i. Here is how you can compute the value of i^i = 0.207879576...
- Since e^(ix) = Cos x + i Sin x, then let x = Pi/2.
- Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i.
- Therefore e^(iPi/2) = i.
- Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(-Pi/2).
- Therefore i^i = e^(-Pi/2) = .207879576...
Does this seem familiar? 1/2 + i t
and rounding out some stuff; as well as sin - cos 61.5 sin elog10- cos = = -0.992420195199509
0.047425172189568
1c ((xn^[1/2]) + 0/n - (x)) = 0 n = {Integral{log(e)}} n = {Integral{log(e)}} = e(log(e) - 1)
Now I am going to increment i numbers by 1/2, add them, for pooling prime numbers 7.5i 11.5i 13.5i 5675055181i and 5675055181.5i and 56750551851i 1753.5i 17.5i
7.5 + 11.5 is 19, a prime number 17.5 + 11.5 is 29, a prime number
now apply x = sqrt(x^2 + 1)
5,675,055,243 = +
1.442695040888963 = log2(e)
5675055181 is prime, a number i wrote wrongly (567055181...)
sqrt (sin(sin(0))) ln(e) - 1 = -7.6505219853240925714770858378589e-48
normalized vector (0.74664, 0.665228) vector length : 0.7594276 angles between vector and coordinate axis xycostheta horizontal: 41.6998 | vertical 48.300
Addition and count of the prime numbers is important here. Then we get into means, Likelihood function, Laurent series all in order to understand the pattern and parameters we are working with. Let’s initiate a function, as I have been doing. constant q = 8.
- Likelihood function: how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model
To be continued
So, let's begin again with a different goal in mind to the end of this mathematics. With similar rules for beginning equations, I want to then take 4 random (goal is prime numbers) numbers and plot them, base a graph calculus between the two plotted points (square A exists above), and continue into the more finer work below (including the reference of the transcendetnal number 1/log(2)) Just working with e, pi, and fraction/exponents gives a squeezing number line in order to facilitate the _____
https://arxiv.org/pdf/2010.15369 https://scipp.ucsc.edu/~haber/ph222/One%20Loop%20Renormalization%20of%20the%20Electroweak%20SM.pdf
<> 59.99 + (1/log(2)) value calculated = 63.311928094887362347870319429489</> <> 59.99 - (1/log(2)) value calculated = 56.668071905112637652129680570511</> <>
<> base: 56.668071905112637652129680570511 — 63.311928094887362347870319429489 n=/=0 and this constant... This constant 8.0365 , e^-π/2, and 1/log(2) e^-π/2 = i^i = ~.2
1 / (1 - a(p)/(p)) 1 / (1 - 7( e^-π/2)/( e^-π/2)) = 1/-6
How interesting
e^e - pi / 2 = 13.56346...
</> and we will be snaggin this n = {Integral{log(e)}} from earlier = e(log(e) - 1)
square 2: - 7 a - 1/-6 b - e^-π/2 c - 1/log(2) d - e(log(e) - 1)https://mathworld.wolfram.com/PrimePartition.html
q = 1 - p since p − 1 ≡ −1 (mod p) q = -1
-1 not trivial zero equality between a sum and a product 1 sum = product q = -1 q = -1 * n : inf 1 - p = -1 * n (1-p)/n = -1 -1/n - p/n = -11/n + p/n = 1
p - 1 = -1 mod p p = 0
1/n + 0/n = 1 n =/= 0
1/n + 0/n - 1 = 0 1((x/n) + 0/n - (x)) = 0
1 ((xn^[1/2]) + 0/n - (x)) = 0 n = {Integral{log(e)}}
the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2
square 1 a1 :: 7 = ! b1 :: 11 c1 :: 3 d1 :: 13
square 2:
- 7 a
- 1/-6 b
- e^-π/2 c
- 1/log(2) d
- e(log(e) - 1)
ii = 0.207879576... (Here i is the imaginary number sqrt(-1).
sqrt(1/log(2)) = 1.2011224087...
i^i is equal to e(- pi / 2 ) and several other values. Consider i^i = e(i log i ) = e( i times i pi / 2 ) . Since log is multivalued, there are other possible values for i^i.
Here is how you can compute the value of i^i = 0.207879576...
1. Since e^(ix) = Cos x + i Sin x, then let x = Pi/2.
2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i.
3. Therefore e^(iPi/2) = i.
4. Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(-Pi/2).
5. Therefore i^i = e^(-Pi/2) = .207879576...
sqrt(e^(-pi/2)) = 0.4559381277659962367659212947280294194166043652378518699962909794, transcendental
e^(-π/2)/sqrt(2) = 0.1469930581078104003917851214212680172128180592460264957869964830, transcendental
some like sqrt(e^(-pi/2))?
this value is transcendental, but i cant seem to find the equation i had that solved for it.
this is the imaginary part.
1.099035748440769286220104851475676502528692878885110516870721630171012356891058664762006751384705773411555698997354, transcendental
(e^-pi) / 2 = 0.021606959131886124887208868585864005637864054905316541490359843700525382878508983849069979980950542193508403482298, transcendental
e^(-pi/2) = 0.2078795763507619085469556198349787700338778416317696080751358830, transcendental
sqrt i^i = sqrt(e^-pi/2)
sqrt of i^i is i, sqrt of e^-pi/2 is 1.099035...
is i = 1.099035748440769286220104851475676502528692878885110516870721630171012356891058664762006751384705773411555698997354
Does this seem familiar?
1/2 + i t
1c ((xn^[1/2]) + 0/n - (x)) = 0
n = {Integral{log(e)}}
now apply f(x) = sqrt(x^2 + 1)
sqrt(7^2 + 1)
sqrt((1/-6)^2 + 1)
- sqrt((e^-π/2)+1)
- sqrt((1/log(2))+1)
- sqrt((e(log(e) - 1) + 1)
ln(e) - 1 = -7.6505219853240925714770858378589e-48
<h4>## Laurent series</h4>
to be done
A pattern of prime numbers,
- sin(z)/z^n, sin(1/z)
The likelihood function, parameterized by a (possibly multivariate) parameter theta, is usually defined differently for discrete and continuous probability distributions (a more general definition is discussed below). Given a probability density or mass function
x -> f(x | theta)
where x is a realization of the random variable X, the likelihood function is theta -> f(x | theta)
often written L(theta | x)
<h4>### Feynman functional integral </h4>
https://www.wolframalpha.com/input?i2d=true&i=%5C%2840%29Log%5B2%2Ce%5D%5C%2841%29+%2B+%5C%2840%29Log%5Be%5D-1%5C%2841%29
https://www.wolframalpha.com/input?i=%28log2%28e%29%29+%2B+%28ln%28e%29+-+1%29
<h4>(log2(e)) + (ln(e) - 1) = 1/log(2) or log2(e)?</h4>
<h3>1 / log(2) is a transcendental number</h3>
My conclusions,
Complex Gravity Analysis
G+ i+L, i imaginary number, Gravity constant G, and length measurement (for quantum probabilities)
Gravity constant + iL(theta | x),
s = 0 + it
<h4>1.442695040888963 = log2(e)</h4>
-0.0450174147692956119367488822382 * ((0.81007883)^2 - (10 * ( 1 / log(2)^2)) =
= 11.035206267601980662683842299568
== -109.69583496520563772683842299568
4.9382229011404647012246185768126
0.97027011439203392574025601921001
<h2>continued fraction theorem</h2>
Let x be an irrational number where 0<x<1 and
d_n (x) = 10^(-n) floor(10^n x)
e_n (x) = 10^(-n) (floor(10^n x) + 1)
be decimal approximations of x, m be a Lebesgue measure set, x = continued fraction k _(n=1)^∞ 1/a_n
be the regular continued fraction of x, d_n(x) = continued fraction k _(n=1)^∞ 1/(b_1(n))
be the regular continued fraction of d_n(x), e_n(x) = continued fraction k _(n=1)^∞ 1/(b_2(n))
be the regular continued fraction of e_n(x), and
k_n(x) = sup({i: for all i<=n, b_1(i) = b_2(i)}).
Then
for almost all x, lim_(n->∞) k_n/n = (6ln 2 ln 10)/π^2.
https://scipp.ucsc.edu/~haber/ph222/One%20Loop%20Renormalization%20of%20the%20Electroweak%20SM.pdf
Assume that X
is a finite-dimensional Banach space. I know that, in general, if two functions f,g:X→R
are convex, then the function (f−g):X→R
given by x↦f(x)−g(x)
is not necessarily convex. Are there conditions we can impose on f
and g
so that the difference is still convex, e.g., if f(x)≥g(x)
https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory#Dynamic_beam_equation
<h4>Sources:</h4>
- Quantum Field Theory I: Basics in Mathematics and Physics Zeidler, Eberhard 2006/01/01 - Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists
- https://scholar.google.com/citations?view_op=view_citation&hl=en&user=4wpjDroAAAAJ&citation_for_view=4wpjDroAAAAJ:u5HHmVD_uO8C
- [wolfram alpha](https://www.wolframalpha.com/input?i=%7B.567055181%2C+.5052250171%7D+graph)
- https://math.stackexchange.com/questions/959339/when-is-the-difference-of-two-convex-functions-convex
- https://en.wikipedia.org/wiki/Euler_product
- https://en.wikipedia.org/wiki/Partition_of_a_set
- https://mathworld.wolfram.com/PrimePartition.html
- https://en.wikipedia.org/wiki/Carmichael_number
- https://en.wikipedia.org/wiki/Agoh%E2%80%93Giuga_conjecture
- https://en.wikipedia.org/wiki/Giuga_number
- https://en.wikipedia.org/wiki/Laurent_series
- https://en.wikipedia.org/wiki/Power_series
- https://en.wikipedia.org/wiki/Holomorphic_function
- https://en.wikipedia.org/wiki/Green%27s_theorem
- https://en.wikipedia.org/wiki/Stokes%27_theorem
- https://sprott.physics.wisc.edu/pickover/trans.html
- https://scipp.ucsc.edu/~haber/ph222/One%20Loop%20Renormalization%20of%20the%20Electroweak%20SM.pdf
https://www.physicsforums.com/threads/what-does-a-negative-or-imaginary-partition-function-indicate.752621/
](https://github.com/gabrielc42/q-and1-p-characters-of-Prime-numbers/blob/main/stuff.md
How Fish see the world...
<h1># Characteristics of numbers 11, 7, 3, 5675055181, 13, 1753, 56750551851, 17, 29, 19, 2, 5, 4273, 127, 23 and other transcendental numbers
under Bernoulli distribution of influence of sets and Likelihood function,
to discuss physical laws of prime numbers, imaginary partition functions, and the Riemann hypothesis.
Continued reflection on proof by Laurent series and Likelihood function.
Latter conjecture about Electroweak fields, fluid dynamics, and holomorphism.
</h1>
<h4>In this discussion I present a series of equatins, arithmetic, and various utilities in order to studying a multitude of systems. </h4>
I begin with pertitent information regarding origin points, but in reality I began with just numbers.
Continuous reasoning and findings of prime numbers alongside partition review led me to discover for a another time the significance of prime numbers and partitions, including the zeta function.
<h4>I brought about several research topics that deal with physical laws and applications between these numbers and characteristics, as well as methodologies simply derived and interpreted by myself
and hopefully following some rational sound math formula and strategies in order to rediscover through a similar path an array of transcendental numbers,
including a lost one, dealing with other various intricacies and multiplots of prime numbers, special formulas and conjectures, all in order to study systems that exhibit quantum mechanical behavior,
such as some particles. Imaginary thermodynamics anyone?</h4>
<h4>In this paper , https://link.springer.com/chapter/10.1007/978-3-642-87140-5_2</h4>
the proof of the important Orlicz- Pettis theorem which claims the equivalence of weak and strong unconditional convergence of series in Banach spaces.
The second paragraph contains Riemann’s theorem which asserts that absolute and unconditional convergence of series in finite dimensional vector spaces are the same,
and the famous Dvoretzky-Rogers theorem. The latter states the existence in every infinite dimensional Banach space of an unconditional series which is not absolutely convergent,
a fact, which has been conjectured for about twenty years and which has been settled down by Dvoretzky and Rogers in 1950.
<h4>In this PHD dissertation, Convergence of Frame Series from Hilbert spaces to Banach spaces And l^1-boundedness, https://repository.gatech.edu/entities/publication/36f42a49-2e32-4cb4-9f3e-a6196c623a47,
similar methods and information are mentioned.</h4>
<H4><Calculation of Gibbs partition function with imaginary time evolution on near-term
quantum computers </H4>
https://arxiv.org/pdf/2109.14880
<h4>Liouville's theorem (complex analysis) states that every bounded entire function must be constant.</h4>
<> 59.99 value calculated </>
<h4><> $`cos(pi/2 - log2(e)) = 0.99999750062433397480084412018728`$ </></h4>
<> == 99999750062433397480084412018728, not a prime </>
<> $`60sin(3.659261913760489 theta) = 3.829365617345466730201708252119
`$
</>
<> == 3659261913760489 -> is divisible by **23**, a *prime number* </>
<> == 3829365617345466730201708252119 is not a prime </>
<H4><> base: 59.999 —</H4>
n=/=0
and this constant...
This constant 8.0365 ,
1 / (1 - a(p)/(p))
1 / (1 - 23(8.0365)/(8.0365)) = -0.04545454545454545454545454545455
Graph this, {.567055181, .5052250171} graph in [wolfram alpha](https://www.wolframalpha.com/input?i=%7B.567055181%2C+.5052250171%7D+graph),
</>
<h2> ## Number(Prime count): </h2>
- 11 (
- 7
- 3,
- 5675055181
- 13
- 1753
- 56750551851
- 17
- 29 (1)
- 19 (1)
- 2 (1)
- 5 (1)
- 4273
- 127
- 23 (1)
<h1>### What do these numbers have in common?</h1>
14 dimensions, prime, their dreams
<h3>### Entropy,</h3>
<> $`H(X) = Epln(1/P(X)) = -[P(X = 0)lnP(X=0)+P(X=1)lnP(X=1)]`$ </>
<> $`H(X) = -(qlnq+plnp), q = P(X=0), p = P(X = 1)`$ </>
<> The entropy is maximized when p = 0.5, indicating the highest level of uncertainty when both outcomes are equally likely. The entropy is zero when or p=0, or p = 1., where one outcome is certain. </>
https://mathworld.wolfram.com/PrimePartition.html
<h4>
``
*key words:
radio waves, partitions, prime numbers characteristics, Riemann hypothesis, Bernoulli distribution, Likelihood function, Euler product, meromorphic function, sets, summation, Laurent series, addition, series, pattern, gravity, imaginary numbers, complex analysis, isolated singularities, entropy, Agoh–Giuga conjecture, Euler product, Banach space, Prime partition, Guiga number, Carmichael number, number theory, topological quantum matter, manifolds, Fourier transformation, Laplace transformation, complex differentiables, imaginary numbers, Gauss equation, Ramanujan, Euler, imaginary partition function, transcendental numbers
*
``
</h4>
970,462.32026163842254755194171738
95426903.18473885
In the 19th century, physicists developed the methods of statistical mechanics for studying many-particle systems, whereas mathematicians proved the distribution law for prime numbers. It turns out that the two apparently different approaches can be traced back to the same mathematical root, namely, the notion of partition function. In modern quantum field theory, the Feynman functional integral can be viewed as a partition function, as we will discuss later on. The typical procedure proceeds in the following two steps.
The imaginary partition function is useful in studying systems that exhibit quantum mechanical behavior, such as atoms and molecules
Reference: https://www.physicsforums.com/threads/what-does-a-negative-or-imaginary-partition-function-indicate.752621/
<h4>It's my lucky day. I solved for prime numbers. I was messing around with equations and papers and homework and math. Got into prime numbers based off ideas of 10 digit numbers and divisibility, and into Bernoulli’s distribution. I went through [partitions](
https://scholar.google.com/citations?view_op=view_citation&hl=en&user=4wpjDroAAAAJ&citation_for_view=4wpjDroAAAAJ:u5HHmVD_uO8C), into [Bernoullis distribution](https://en.wikipedia.org/wiki/Bernoulli_distribution), and [numbers](https://en.wikipedia.org/wiki/Bernoulli_number), from [Fermat’s little theorem](https://en.wikipedia.org/wiki/Proofs_of_Fermat's_little_theorem), and unto of course my flavor into the interpretations and sudo math., inlcuding...
</h4>
q = 1 - p
since
p − 1 ≡ −1 (mod p)
q = -1
<h4> </h4>
-1 not trivial zero
equality between a sum and a product
<h4> </h4>
1 sum = product
q = -1
q = -1 * n : inf
1 - p = -1 * n
(1-p)/n = -1
-1/n - p/n = -1
1/n + p/n = 1
p - 1 = -1 mod p
p = 0
1/n + 0/n = 1
n =/= 0
1/n + 0/n - 1 = 0
1((x/n) + 0/n - (x)) = 0
1 ((xn^[1/2]) + 0/n - (x)) = 0
n = {Integral{log(e)}}
<h4>the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2</h4>
x = sqrt(x^2 + 1)
x = {0, |x| - 1}
x = {0, (|log(e)| - 1) * } = {0, .5657055181}
{.567055181, .5052250171}
{b,c} graph
b is divisible by 7, c is divisible by 11
11 and 7 are prime numbers
11 and 7 at integer magnitude = b and c
b / 7 = 81,007,883
refract at magnitude change: .81007883
c / 11 = 205,004,561 : 2.05004561
<H4>remember |log(e)| - 1 = .5675055181 </H4>
.56570055 = 56570055 is divisible by 3
.5675955181 = 5675955181 is divisible by 13
.567055181 = 567055181 is divisible by 7
x = log(e)
e * pi = 8.5397
.81007883 + .5675055181 =
8.5397 - .5675055181 = 7.972287 , 8.1007883 == 8.0365
——————————
<h4>This constant 8.0365 will be valuable later on. It is a base off our n=/=0 earlier. Providing at least one value is all we need in order to understand what the root of the partition, n, is. This will explain the patterns, physical applications, and unique prime characteristics that follow the 19th century in which physicists developed the methods of statistical mechanics for studying many-particle systems, whereas mathematicians proved the distribution law for prime numbers. It turns out that the two apparently different approaches can be traced back to the same mathematical root, namely, the notion of partition function. In modern quantum field theory, the Feynman functional integral can be viewed as a partition function, as we will discuss later on.</h4>
https://en.wikipedia.org/wiki/Partition_of_a_set
<H4>Characteristics of prime numbers 11, 7, 3, 5675055181, 13, 1753, 56750551851, 17, 29, 2, 5, 4273, 127</H4>
https://socratic.org/questions/587d0569b72cff585cd949
integral(sqrt(x^2 + 1)dx) = (ln|x+(sqrt(x^2 + 1))| + x(sqrt(x^2 + 1)))/2 + c
= sqrt(x^2 + 1)
x = sqrt(x^2 + 1)
x =11, 7, … any other prime numbers?
Is what is below similar to what is above?
It needs to be proved.
11 >> 11.04536101718726 == 1104536101718726 , is divisible by 2 a *prime number*
7 >> 7.071067811865475 == 7071067811865475 is divisible by 5 a *prime number*
Occams razor, probabilistic determination here.
<H4>### What is divisible by the prime numbers?</H4>
<h4>567055181 is divisible by 7, a *prime number*</h4>
<H4>81007883 is divisible by 11, a *prime number*</H4>
<H4>7972287, 56570055, 56750551851, 5,675,055,243 is divisible by 3, *a prime number*</H4>
<H4>5675955181 is divisible by 13, a *prime number*</H4>
<H4>1442695040888963 is divisible by 17, a *prime number*</H4>
<H4>7071067811865475 is divisible by 5, a *prime number*</H4>
<H4>56750551851 is *prime*</H4>
<H4>29 is *prime*</H4>
<H4>7,364,353 is divisible by 1753, a *prime number*</H4>
i^i is equal to e(- pi / 2 ) and several other values. Consider i^i = e(i log i ) = e( i times i pi / 2 ) . Since log is multivalued, there are other possible values for i^i. Here is how you can compute the value of i^i = 0.207879576...
2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i.
4. Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(-Pi/2).
Does this seem familiar?
and rounding out some stuff; as well as sin - cos 61.5 sin elog10- cos = = -0.992420195199509
0.047425172189568
1c ((xn^[1/2]) + 0/n - (x)) = 0 n = {Integral{log(e)}} n = {Integral{log(e)}} = e(log(e) - 1)
7.5i 11.5i 13.5i 5675055181i and 5675055181.5i and 56750551851i 1753.5i 17.5i5,675,055,243 = +
1.442695040888963 = log2(e)
sqrt (sin(sin(0)))
normalized vector (0.74664, 0.665228) vector length : 0.7594276 angles between vector and coordinate axis xycostheta horizontal: 41.6998 | vertical 48.300
Addition and count of the prime numbers is important here. Then we get into means, Likelihood function, Laurent series all in order to understand the pattern and parameters we are working with. Let’s initiate a function, as I have been doing. constant q = 8.
- Likelihood function: how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model
https://arxiv.org/pdf/2010.15369 https://scipp.ucsc.edu/~haber/ph222/One%20Loop%20Renormalization%20of%20the%20Electroweak%20SM.pdf
<> n=/=0 and this constant...How interesting
</>
square 2:https://mathworld.wolfram.com/PrimePartition.html
q = 1 - p since p − 1 ≡ −1 (mod p) q = -1
-1 not trivial zero equality between a sum and a product 1 sum = product q = -1 q = -1 * n : inf 1 - p = -1 * n (1-p)/n = -1 -1/n - p/n = -11/n + p/n = 1
p - 1 = -1 mod p p = 0
1/n + 0/n = 1 n =/= 0
1/n + 0/n - 1 = 0 1((x/n) + 0/n - (x)) = 0
1 ((xn^[1/2]) + 0/n - (x)) = 0 n = {Integral{log(e)}}
a1 :: 7 = ! b1 :: 11 c1 :: 3 d1 :: 13<h4>square 2:</h4>
- 7 a
- 1/-6 b
- e^-π/2 c
- 1/log(2) d
- e(log(e) - 1)
ii = 0.207879576... (Here i is the imaginary number sqrt(-1).
<h4>sqrt(1/log(2)) = 1.2011224087...</h4>
i^i is equal to e(- pi / 2 ) and several other values. Consider i^i = e(i log i ) = e( i times i pi / 2 ) . Since log is multivalued, there are other possible values for i^i.
Here is how you can compute the value of i^i = 0.207879576...
1. Since e^(ix) = Cos x + i Sin x, then let x = Pi/2.
2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i.
3. Therefore e^(iPi/2) = i.
4. Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(-Pi/2).
5. Therefore i^i = e^(-Pi/2) = .207879576...
<h4>sqrt(e^(-pi/2)) = 0.4559381277659962367659212947280294194166043652378518699962909794, transcendental</h4>
<h4>e^(-π/2)/sqrt(2) = 0.1469930581078104003917851214212680172128180592460264957869964830, transcendental</h4>
some like sqrt(e^(-pi/2))?
this value is transcendental, but i cant seem to find the equation i had that solved for it.
this is the imaginary part.
<h4>1.099035748440769286220104851475676502528692878885110516870721630171012356891058664762006751384705773411555698997354, transcendental</h4>
<h4>(e^-pi) / 2 = 0.021606959131886124887208868585864005637864054905316541490359843700525382878508983849069979980950542193508403482298, transcendental</h4>
<h4>e^(-pi/2) = 0.2078795763507619085469556198349787700338778416317696080751358830, transcendental</h4>
<h4>sqrt i^i = sqrt(e^-pi/2)</h4>
sqrt of i^i is i, sqrt of e^-pi/2 is 1.099035...
<h4>is i = 1.099035748440769286220104851475676502528692878885110516870721630171012356891058664762006751384705773411555698997354?</h4>
Does this seem familiar?
1/2 + i t
1c ((xn^[1/2]) + 0/n - (x)) = 0
n = {Integral{log(e)}}
now apply f(x) = sqrt(x^2 + 1)
sqrt(7^2 + 1)
sqrt((1/-6)^2 + 1)
- sqrt((e^-π/2)+1)
- sqrt((1/log(2))+1)
- sqrt((e(log(e) - 1) + 1)
ln(e) - 1 = -7.6505219853240925714770858378589e-48
<h4>## Laurent series</h4>
to be done
A pattern of prime numbers,
- sin(z)/z^n, sin(1/z)
The likelihood function, parameterized by a (possibly multivariate) parameter theta, is usually defined differently for discrete and continuous probability distributions (a more general definition is discussed below). Given a probability density or mass function
x -> f(x | theta)
where x is a realization of the random variable X, the likelihood function is theta -> f(x | theta)
often written L(theta | x)
<h4>### Feynman functional integral </h4>
https://www.wolframalpha.com/input?i2d=true&i=%5C%2840%29Log%5B2%2Ce%5D%5C%2841%29+%2B+%5C%2840%29Log%5Be%5D-1%5C%2841%29
https://www.wolframalpha.com/input?i=%28log2%28e%29%29+%2B+%28ln%28e%29+-+1%29
<h4>(log2(e)) + (ln(e) - 1) = 1/log(2) or log2(e)?</h4>
<h3>1 / log(2) is a transcendental number</h3>
<H4>16 ^ (1/log(2)) = 10,000</H4>
<h4>10 ^ (1/log(2)) = 2098.59239586666...</h4>
<h4>9 ^ (1/log(2)) = 1478.8529821647...</h4>
<h4>8 ^ (1/log(2)) = 1000</h4>
<h4>7 ^ (1/log(2)) = 641.73381175...</h4>
<h4>6 ^ (1/log(2)) = 384.55857579...</h4>
<h4>5 ^ (1/log(2)) = 209.859239...</h4>
<h4>4 ^ (1/log(2)) = 100</h4>
<h4>3 ^ (1/log(2)) = 38.45587...</h4>
<h4>2 ^ (1/log(2)) = 10</h4>
<h4>1 ^ (1/log(2)) = 1</h4>
Neat...however incorrect.
Not sure why my calculator gave wrong answers.
But still,
<h4>2^(1/log(2)) = e</h4>
<h4>16^(1/log(2)) = e^4</h4>
<h4>My conclusions, </h4>
Complex Gravity Analysis
G+ i+L, i imaginary number, Gravity constant G, and length measurement (for quantum probabilities)
Gravity constant + iL(theta | x),
s = 0 + it
<h4>1.442695040888963 = log2(e)</h4>
-0.0450174147692956119367488822382 * ((0.81007883)^2 - (10 * ( 1 / log(2)^2)) =
= 11.035206267601980662683842299568
== -109.69583496520563772683842299568
4.9382229011404647012246185768126
0.97027011439203392574025601921001
<h2>continued fraction theorem</h2>
Let x be an irrational number where 0<x<1 and
d_n (x) = 10^(-n) floor(10^n x)
e_n (x) = 10^(-n) (floor(10^n x) + 1)
be decimal approximations of x, m be a Lebesgue measure set, x = continued fraction k _(n=1)^∞ 1/a_n
be the regular continued fraction of x, d_n(x) = continued fraction k _(n=1)^∞ 1/(b_1(n))
be the regular continued fraction of d_n(x), e_n(x) = continued fraction k _(n=1)^∞ 1/(b_2(n))
be the regular continued fraction of e_n(x), and
k_n(x) = sup({i: for all i<=n, b_1(i) = b_2(i)}).
Then
for almost all x, lim_(n->∞) k_n/n = (6ln 2 ln 10)/π^2.
https://scipp.ucsc.edu/~haber/ph222/One%20Loop%20Renormalization%20of%20the%20Electroweak%20SM.pdf
Assume that X
is a finite-dimensional Banach space. I know that, in general, if two functions f,g:X→R
are convex, then the function (f−g):X→R
given by x↦f(x)−g(x)
is not necessarily convex. Are there conditions we can impose on f
and g
so that the difference is still convex, e.g., if f(x)≥g(x)
https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory#Dynamic_beam_equation
<h4>Sources:</h4>
- Quantum Field Theory I: Basics in Mathematics and Physics Zeidler, Eberhard 2006/01/01 - Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists
- https://scholar.google.com/citations?view_op=view_citation&hl=en&user=4wpjDroAAAAJ&citation_for_view=4wpjDroAAAAJ:u5HHmVD_uO8C
- [wolfram alpha](https://www.wolframalpha.com/input?i=%7B.567055181%2C+.5052250171%7D+graph)
- https://math.stackexchange.com/questions/959339/when-is-the-difference-of-two-convex-functions-convex
- https://en.wikipedia.org/wiki/Euler_product
- https://en.wikipedia.org/wiki/Partition_of_a_set
- https://mathworld.wolfram.com/PrimePartition.html
- https://en.wikipedia.org/wiki/Carmichael_number
- https://en.wikipedia.org/wiki/Agoh%E2%80%93Giuga_conjecture
- https://en.wikipedia.org/wiki/Giuga_number
- https://en.wikipedia.org/wiki/Laurent_series
- https://en.wikipedia.org/wiki/Power_series
- https://en.wikipedia.org/wiki/Holomorphic_function
- https://en.wikipedia.org/wiki/Green%27s_theorem
- https://en.wikipedia.org/wiki/Stokes%27_theorem
- https://sprott.physics.wisc.edu/pickover/trans.html
- https://scipp.ucsc.edu/~haber/ph222/One%20Loop%20Renormalization%20of%20the%20Electroweak%20SM.pdf
https://www.physicsforums.com/threads/what-does-a-negative-or-imaginary-partition-function-indicate.752621/)