eulerTotient.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# How to determine the horizon of the visible universe in a Jupyter Notebook"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 15 correlations establishing the theoritical value of the horizon of the visible universe see [Observable_universe](https://en.wikipedia.org/wiki/Observable_universe) and [Schwarzschild_radius](https://en.wikipedia.org/wiki/Schwarzschild_radius)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "with ± ... billion (10^9) years approximation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "List of physical (NIST CODATA2018) and mathematical constants used:\n",
    "\n",
    "* $\\pi=3.141592653589793$\n",
    "* Euler natural logarithm $e=2.718281828459045$\n",
    "* Euler Mascheroni $\\gamma=0.5772156649015329$\n",
    "* Atiyah's $\\Gamma=25.178097241906$\n",
    "* Lucas Mersenne number [Edouard Lucas](https://en.wikipedia.org/wiki/%C3%89douard_Lucas)\n",
    "* OM Monster [Monster_group](https://en.wikipedia.org/wiki/Monster_group) also known as Fischer–Griess monster\n",
    "* OB Baby Monster [Baby_monster_group](https://en.wikipedia.org/wiki/Baby_monster_group) baby monster group B\n",
    "* Eddington Electric Constant $a=137.0359990836958$ also known as the inversed fine structure constant CODATA2018\n",
    "* $c=299792458.0$ m/s CODATA2018\n",
    "* $h=6.62607015.10^{-34}$ $J.Hz^{-1}$ Planck constant CODATA2018\n",
    "* $\\hbar=1.0545718176461565.10^{-34}$ $J.s$ Planck constant over 2 pi CODATA2018\n",
    "* $l_P=1.616255.10^{-35}$ m Planck length CODATA2018\n",
    "* $m_P=2.176434.10^{-8}$ kg Planck mass CODATA2018\n",
    "* ƛ$_e$ $=3.861 592 6796.10^{-13}$m Reduced (Electron) Compton Wavelength CODATA2018\n",
    "* ƛ$_p$ $=2.103 089 103 36.10^{-16}$m Reduced (Proton) Compton Wavelength CODATA2018\n",
    "* $\\lambda_{wi}=2.897771955.10^{-3}$ m K (exact) Wien wavelength displacement law constant CODATA2018\n",
    "* $k_B=1.380 649.10^{-23}$ J K^-1  (exact) Boltzmann constant CODATA2018\n",
    "* $t_K=9600.60(1)$s non-Doppler Kotov Periodicity (1000 ppb)\n",
    "* Fermi coupling constant : $1.1663787.10^{-5}$    (GeV)-2          (51 ppb)\n",
    "* Fermi's ratio   F = Fermi's mass / m_e = 573007.3625        (25 ppb)\n",
    "* Fermi-Atiyah's ratio : $F_A = (2\\Gamma . 137)^{3/2}$ ≈ 573007.3652        (0.22 ppb)\n",
    "* Mass of the electron $m_e=9.1093837015.10^{-31}$ kg CODATA2018\n",
    "* Mass of the proton $m_p=1.672 621 923 69.10^{-27}$ kg CODATA2018\n",
    "* Magnetic Moment of the electron/Bohr magneton: $d_e$ ≈ 1.00115965218128 (2 × 10-4  ppb)\n",
    "* Magnetic moment anomaly $1159.65218091.10^{−6}$ ± 0.00000026 [Particle Data Group Leptons](http://pdg.lbl.gov/2019/tables/rpp2019-sum-leptons.pdf) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019\n",
    "* Boson $W=80.379 GeV$ ± 0.012 [Particle Data Group Bosons](http://pdg.lbl.gov/2019/tables/rpp2019-sum-gauge-higgs-bosons.pdf) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019\n",
    "* Boson $Z=91.1876 GeV$ ± 0.0023 [Particle Data Group Bosons](http://pdg.lbl.gov/2019/tables/rpp2019-sum-gauge-higgs-bosons.pdf) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019\n",
    "* Lepton $e=0.5109989461 MeV$ [Particle Data Group Leptons](http://pdg.lbl.gov/2019/tables/rpp2019-sum-leptons.pdf) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019\n",
    "* Baryon $p=938.272081 MeV$ [Particle Data Group Baryons](http://pdg.lbl.gov/2019/tables/rpp2019-sum-baryons.pdf) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019\n",
    "* Baryon $n=939.565413 MeV$ [Particle Data Group Baryons](http://pdg.lbl.gov/2019/tables/rpp2019-sum-baryons.pdf) M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019\n",
    "* $G=6.6743.10^{-11}$ $m^{3}.kg^{-1}.s^{-2}$ Newtonian constant of gravitation CODATA2018\n",
    "* $G_q=6.6755.10^{-11}$ $m^{3}.kg^{-1}.s^{-2}$ Newtonian constant of gravitation measured by T.Quinn et al. (2013) BIPM Sevres [Improved determination of G using two methods](https://www.bipm.org/utils/en/pdf/PhysRevLett.111.101102.pdf)\n",
    "* $G_s=6.67545372.10^{-11}$ $m^{3}.kg^{-1}.s^{-2}$ Newtonian constant of gravitation estimate by Francis M. Sanchez (Jan 2020)\n",
    "* Galaxies Doppler radius $R=c.f /\\Delta.f$=Universe Schwarzschild radius: $2G.M/c^2$\n",
    "* Hubble-Lemaître radius: $c/H_0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "170141183460469231731687303715884105727\n",
      "808017424794512875886459904961710757005754368000000000\n",
      "4154781481226426191177580544000000\n"
     ]
    }
   ],
   "source": [
    "Lucas=(2**127)-1\n",
    "print(Lucas)\n",
    "OM=2**46 * 3**20 * 5**9 * 7**6 * 11**2 * 13**3 * 17 * 19 * 23 * 29 * 31 * 41 * 47 * 59 * 71 # 808017424794512875886459904961710757005754368000000000\n",
    "print(OM)\n",
    "OB=2**41 * 3**13 * 5**6 * 7**2 * 11 * 13 * 17 * 19 * 23 * 31 * 47 # 4154781481226426191177580544000000\n",
    "print(OB)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 3**210"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.67545372e-11\n"
     ]
    }
   ],
   "source": [
    "# Using G_s the Newtonian constant of gravitation estimate by Francis M. Sanchez (Jan 2020)\n",
    "G_s=6.67545372*10**-11\n",
    "print(G_s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import constants\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "299792458.0 6.62607015e-34 1.0545718176461565e-34 137.0359990836958\n"
     ]
    }
   ],
   "source": [
    "c=constants.c\n",
    "h=constants.h\n",
    "hbar=constants.hbar\n",
    "a=137.0359990836958\n",
    "print(c,h,hbar,a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.718281828459045 0.5772156649015329 25.178097241906\n"
     ]
    }
   ],
   "source": [
    "e=np.e\n",
    "gamma=np.euler_gamma\n",
    "Gamma=(a*gamma)/np.pi\n",
    "print(e,gamma, Gamma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "573007.364905975\n"
     ]
    }
   ],
   "source": [
    "# F**(2/3)=2*137*Gamma\n",
    "# Fermi's mass given by \n",
    "F=(2*137*Gamma)**1.5\n",
    "print(F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.8615926796089057e-13\n",
      "6.739167620022749e-19\n",
      "2.1030891033555923e-16\n"
     ]
    }
   ],
   "source": [
    "lambdabare=constants.hbar/(constants.m_e*constants.c)\n",
    "print(lambdabare)\n",
    "lambdabar_F=lambdabare/F\n",
    "print(lambdabar_F)\n",
    "lambdabar_p=(constants.hbar/(constants.m_p*constants.c))\n",
    "print(lambdabar_p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {},
   "outputs": [],
   "source": [
    "m_e=constants.m_e\n",
    "m_p=constants.m_p\n",
    "m_n=constants.m_n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1837.152646 1836.15267343 1838.68366173\n"
     ]
    }
   ],
   "source": [
    "H=1837.152646   # (0.06 ppb)\n",
    "p=1836.15267343 # 1836.1526734400013 #(0.06 ppb)\n",
    "n=1838.68366173 # 1838.6836617324586 #(0.5 ppb)\n",
    "print(H,p,n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Universe Horizon Radius calculus in 3 minutes (method 1)\n",
    "$$R_U=\\frac{2.\\hbar^2}{G.m_e.m_p.m_n}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3056258355517771e+26\n"
     ]
    }
   ],
   "source": [
    "R_U=2*constants.hbar**2/(G_s*constants.m_e*constants.m_p*constants.m_n)\n",
    "print(R_U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Yields 13.802861522337478 Gly (G) and 13.80047597102314 Gyr (G_s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.80047597102314"
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_U/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Nambu Universe Radius calculus in 3 minutes (method 2)\n",
    "$$R_N=\\frac{2.\\hbar^2}{G.(m_N)^3}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### https://en.wikipedia.org/wiki/Yoichiro_Nambu"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With Nambu mass $$m_N=m_e.a$$ and $$a=1/\\alpha$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "137.0359990836958\n"
     ]
    }
   ],
   "source": [
    "a=(1/constants.fine_structure)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.2483134965717876e-28\n"
     ]
    }
   ],
   "source": [
    "m_N=constants.m_e*a\n",
    "print(m_N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.7128941663434102e+26\n"
     ]
    }
   ],
   "source": [
    "R_N=2*constants.hbar**2/(G_s*((a)*constants.m_e)**3)\n",
    "print(R_N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One obtains 18.10530562420883 Gly with method 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "18.10530562420883"
      ]
     },
     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_N/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$R_N'=ƛ_e(2R/R_N)^{210}$$ and $$R=\\frac{2.\\hbar^2}{G.m_e.m_p.m_H}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$H=p+1-\\frac{1/2}{a'^2}$$ and $a'=H/p=137.11$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.67262192369e-27"
      ]
     },
     "execution_count": 159,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# proton mass 1.672 621 923 69 e-27\n",
    "constants.m_p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.6735328361530618e-27\n"
     ]
    }
   ],
   "source": [
    "H=constants.m_e*1837.152645\n",
    "print(H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3067138970179814e+26\n"
     ]
    }
   ],
   "source": [
    "R=2*constants.hbar**2/(G_s*constants.m_e*constants.m_p*H)\n",
    "print(R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3108749648748815e+26\n"
     ]
    }
   ],
   "source": [
    "R_Nprime=(constants.hbar/(constants.m_e*constants.c))*(2*R/R_N)**210\n",
    "print(R_Nprime)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One obtains 13.855959311747384 Gly with method 2'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.855959311747384"
      ]
     },
     "execution_count": 163,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_Nprime/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Eddington Universe Radius calculus in 3 minutes (method 3)\n",
    "$$R_{Edd}=\\frac{2.\\hbar^2}{G.m_e.m_p^2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### https://en.wikipedia.org/wiki/Eddington_number"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "15747724136275002577605653961181555468044717914527116709366231425076185631031296\n"
     ]
    }
   ],
   "source": [
    "# Number of Hydrogen Atoms of mass m_H in the visible universe\n",
    "N_Edd=136*2**256\n",
    "print(N_Edd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.6735328361530618e-27\n"
     ]
    }
   ],
   "source": [
    "m_H=1837.152645*constants.m_e\n",
    "print(m_H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.666666666666667"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "20/3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [],
   "source": [
    "R_Edd=(6.666666666666667*N_Edd*G_s*m_H)/constants.c**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One obtains 13.793532128031092 Gly for R_Edd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.793532128031092"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_Edd/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3049688972818604e+26\n"
     ]
    }
   ],
   "source": [
    "R_Edd2=2*constants.hbar**2/(G_s*constants.m_e*constants.m_p**2)\n",
    "print(R_Edd)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One obtains 13.819498813523245 Gly for R_Edd2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.819498813523245"
      ]
     },
     "execution_count": 170,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_Edd2/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Francis M. Sanchez Universe Radius calculus in 3 minutes (method 4)\n",
    "$$R_{Francis.M.Sanchez}=ƛ_e.g(6)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With $$g(k)=exp(2^{k+1/2})/k$$ when $$k=6$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and ƛ$_e=\\frac{\\hbar}{m_e.c}$ as the reduced Compton (electron) wavelength"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.8615926796089057e-13"
      ]
     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Reduced Compton Wavelength\n",
    "(constants.hbar/(constants.m_e*constants.c))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.307568892158173e+26\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy import constants\n",
    "R_FMS=(constants.hbar/(constants.m_e*constants.c))*np.exp(2**(6+(1/2)))/6\n",
    "print(R_FMS)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One obtains 13.821014095558315 Gly with method 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.821014095558315"
      ]
     },
     "execution_count": 173,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_FMS/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Universe Radius calculus based on Rydberg reduced wavelength (method 5) \n",
    "$$R_{Rydberg}=\\frac{(ƛ_{Ryd}.n^4)^2}{ƛ_p}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "137.0359990836958\n"
     ]
    }
   ],
   "source": [
    "a=constants.fine_structure**-1\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "18778.865044866678"
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1838.6836617324586\n"
     ]
    }
   ],
   "source": [
    "# n equals mass ratio between Neutron and Electron\n",
    "n=(constants.m_n)/(constants.m_e)\n",
    "print(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11429522003521.688"
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n**4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.4503265557724145e-08\n"
     ]
    }
   ],
   "source": [
    "lambdabarRyd=2*(constants.hbar/(constants.m_e*constants.c))*(a**2)\n",
    "print(lambdabarRyd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.1030891033555923e-16\n"
     ]
    }
   ],
   "source": [
    "lambdabar_p=(constants.hbar/(constants.m_p*constants.c))\n",
    "print(lambdabar_p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3065621143319103e+26\n"
     ]
    }
   ],
   "source": [
    "R_Ryd=((lambdabarRyd*n**4)**2)/lambdabar_p\n",
    "print(R_Ryd)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One obtains 13.810372445537942 Gly with method 5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.810372445537942"
      ]
     },
     "execution_count": 181,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Incorrect result , method to be modified\n",
    "R_Ryd/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Universe Radius calculus based on pi (method 6) \n",
    "$$R_{\\pi}=ƛ_e.\\pi^{155/2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3058148293550039e+26\n"
     ]
    }
   ],
   "source": [
    "R_pi=(constants.hbar/(constants.m_e*constants.c))*constants.pi**(155/2)\n",
    "print(R_pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One obtains 13.802473637099501 Gly with method 6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.802473637099501"
      ]
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_pi/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Universe Radius calculus based on 2 (method 7) \n",
    "$$R_{2}=ƛ_e.2^{128}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3140318971018876e+26\n"
     ]
    }
   ],
   "source": [
    "R_2=(constants.hbar/(constants.m_e*constants.c))*2**128\n",
    "print(R_2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One obtains 13.889328111716429 Gly with method 7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.889328111716429"
      ]
     },
     "execution_count": 185,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_2/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Universe Radius calculus based on Fermi's mass and Kotov period (method 8) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$R_{kotov-fermi}=\\frac{2.(c.t_k)^2/ƛ_e}{F^2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fermi's mass given by the relation: $$F=(2\\times 137 \\times \\Gamma)^{3/2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "25.178097241906\n"
     ]
    }
   ],
   "source": [
    "# Atiyah's Gamma\n",
    "Gamma=a*np.euler_gamma/np.pi\n",
    "print(Gamma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "573007.364905975\n"
     ]
    }
   ],
   "source": [
    "# F**(2/3)=2*137*Gamma\n",
    "# Fermi's mass given by \n",
    "F=(2*137*Gamma)**1.5\n",
    "print(F)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.3067162206121416e+26"
      ]
     },
     "execution_count": 188,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((2*(constants.c*9600.6)**2)/(constants.hbar/(constants.m_e*constants.c)))/F**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3067162206121416e+26\n"
     ]
    }
   ],
   "source": [
    "R_kf=(2*(constants.c*9600.6)**2)/(constants.hbar/(constants.m_e*constants.c)) / F**2\n",
    "print(R_kf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://en.wikipedia.org/wiki/Light-year"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9460730472580800.0"
      ]
     },
     "execution_count": 190,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# https://en.wikipedia.org/wiki/Light-year\n",
    "constants.light_year"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.812001350205271"
      ]
     },
     "execution_count": 191,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_kf/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Universe Radius calculus based on \"a\"  (method 9) \n",
    "$$a=1/\\alpha$$ $\\alpha$ being the fine structure constant  $$R_{a}=ƛ_e.(2.\\pi^2.a^3)^5$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.8615926796089057e-13\n"
     ]
    }
   ],
   "source": [
    "lambdabare=(constants.hbar/(constants.m_e*constants.c))\n",
    "print(lambdabare)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.2276645699260076e+33"
      ]
     },
     "execution_count": 193,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(2*constants.pi**2)*((1/constants.fine_structure)**3)**5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3059760854303987e+26\n"
     ]
    }
   ],
   "source": [
    "R_a=((constants.pi**2*(2*(1/constants.fine_structure)**3))**5)*(constants.hbar/(constants.m_e*constants.c))\n",
    "print(R_a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.80417811516134"
      ]
     },
     "execution_count": 195,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_a/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Universe Radius calculus based on $CMB$ (method 10) \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$l_P$ is the Planck length , $j$ is the sternheimer constant"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$R_{cmb} = \\frac{(\\lambda_{cmb}/(j+1))^2}{l_P}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "113.91063459990004\n"
     ]
    }
   ],
   "source": [
    "j=(8*np.pi**2)/np.log(2)\n",
    "print(j)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.380649e-23 2.72582\n"
     ]
    }
   ],
   "source": [
    "k_B=constants.Boltzmann\n",
    "T_cmb=2.72582\n",
    "print(k_B,T_cmb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0008400718024707187\n"
     ]
    }
   ],
   "source": [
    "lambdabar_cmb=(constants.hbar*constants.c)/(k_B*T_cmb)\n",
    "print(lambdabar_cmb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.005278326806259892\n"
     ]
    }
   ],
   "source": [
    "lambda_cmb=lambdabar_cmb*2*np.pi\n",
    "print(lambda_cmb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.6163947113699356e-35\n"
     ]
    }
   ],
   "source": [
    "# Planck's length CODATA2018 (hbar.G/c^3)^1/2 = 1.61639471.10^-35\n",
    "# l_P=1.616255*10**(-35) CODATA2018\n",
    "l_P=((constants.hbar*G_s)/constants.c**3)**0.5\n",
    "print(l_P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3053431213288355e+26\n"
     ]
    }
   ],
   "source": [
    "R_cmb=((lambda_cmb/(j+1))**2)/l_P\n",
    "print(R_cmb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.79748767932874"
      ]
     },
     "execution_count": 202,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_cmb/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Universe Radius calculus based on $ƛ_w$, $\\gamma$ and $\\Gamma$ (method 11) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$R_{ƛ_w}=ƛ_w.(\\frac{2^{70}.\\gamma.n^3}{\\Gamma^4})^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ƛ_w=ƛ_e/F^2$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "137.0359990836958\n"
     ]
    }
   ],
   "source": [
    "a=constants.fine_structure**-1\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "25.178097241906\n"
     ]
    }
   ],
   "source": [
    "# Atiyah's Gamma\n",
    "Gamma=a*np.euler_gamma/np.pi\n",
    "print(Gamma)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.141592653589793"
      ]
     },
     "execution_count": 206,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.1761048867371158e-24\n"
     ]
    }
   ],
   "source": [
    "ƛ_w=(constants.hbar/(constants.m_e*constants.c))/F**2\n",
    "print(ƛ_w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.306713911982143e+26\n"
     ]
    }
   ],
   "source": [
    "R_w=(((2**70*np.euler_gamma*n**3)/Gamma**4)**2)*ƛ_w\n",
    "print(R_w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.811976947966931"
      ]
     },
     "execution_count": 209,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_w/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $R =$  ƛe $ g(6)/ (1 + √(137^2+√136)/jn)$ (method 12) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With $$g(k)=exp(2^{k+1/2})/k$$ when $$k=6$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and ƛ$_e=\\frac{\\hbar}{m_e.c}$ as the reduced Compton (electron) wavelength"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3067138950061914e+26\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy import constants\n",
    "R_12=((constants.hbar/(constants.m_e*constants.c))*np.exp(2**(6+(1/2)))/6)/(1+np.sqrt(137**2+np.sqrt(136))/(j*n))\n",
    "print(R_12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.675453730277392e-11\n"
     ]
    }
   ],
   "source": [
    "G_12=2*constants.hbar**2/(R_12*m_e*m_p*m_H)\n",
    "print(G_12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0006543109055848"
      ]
     },
     "execution_count": 212,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1+np.sqrt(137**2+np.sqrt(136))/(j*n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.811976768530982"
      ]
     },
     "execution_count": 213,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_12/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $R =  \\frac{2.ƛe(p.n/H^2).g(5)^2}{ln^2 (2 – 1/j.a^2)} $ (method 13) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With $$g(k)=exp(2^{k+1/2})/k$$ when $$k=5$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and ƛ$_e=\\frac{\\hbar}{m_e.c}$ as the reduced Compton (electron) wavelength"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import constants\n",
    "# R_13=(2*(constants.hbar/(constants.m_e*constants.c))*(p*n/H**2)*(np.exp(2**(5+(1/2)))/5)**2)/(np.log(2-1/(j*a**2)))**2\n",
    "# print(R_13)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {},
   "outputs": [],
   "source": [
    "#R_13/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {},
   "outputs": [],
   "source": [
    "#(2*(constants.hbar/(constants.m_e*constants.c))*(p*n/H**2)*(np.exp(2**(5+(1/2)))/5)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.480452689883273"
      ]
     },
     "execution_count": 217,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(np.log(2-1/(j*a**2)))**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {},
   "outputs": [],
   "source": [
    "#G_13=2*constants.hbar**2/(R_13*m_e*m_p*m_H)\n",
    "#print(G_13)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $R_N(O_M O_B/n_{ph})^2$ (method 14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.8379999999999998e+87\n"
     ]
    }
   ],
   "source": [
    "n_ph=3.838*10**87\n",
    "print(n_ph)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.7128941663434102e+26"
      ]
     },
     "execution_count": 220,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3105645508705246e+26\n"
     ]
    }
   ],
   "source": [
    "R_14=R_N*(OM*OB/n_ph)**2\n",
    "print(R_14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.852678233132401"
      ]
     },
     "execution_count": 222,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_14/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $\\lambda_e OM^{7/10}$ (method 15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 223,
   "metadata": {},
   "outputs": [],
   "source": [
    "lambda_e=lambdabare*2*np.pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 224,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3186843114021136e+26\n"
     ]
    }
   ],
   "source": [
    "#lambda_e*OM**(7/10)\n",
    "R_15=lambda_e*OM**(7/10)\n",
    "print(R_15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.938504169672099"
      ]
     },
     "execution_count": 225,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_15/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $(2\\pi^{32}Pƛe)^2/R_N$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 226,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.389015908333496e+22\n"
     ]
    }
   ],
   "source": [
    "# \n",
    "P=lambdabare/l_P\n",
    "print(P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 227,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3058470789771101e+26\n"
     ]
    }
   ],
   "source": [
    "# Radius in m\n",
    "R_16=((2*np.pi**32)*P*lambdabare)**2/R_N\n",
    "print(R_16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 228,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.802814515874134"
      ]
     },
     "execution_count": 228,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Radius in Gly\n",
    "R_16/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $ƛ_FP^3/p^7$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 229,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.389015908333496e+22"
      ]
     },
     "execution_count": 229,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.30587575365127e+26\n"
     ]
    }
   ],
   "source": [
    "R_17=(lambdabar_F*P**3)/p**7\n",
    "print(R_17)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 231,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.803117607419157"
      ]
     },
     "execution_count": 231,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Radius in Gly\n",
    "R_17/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $ƛ_FP^2e/8$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 232,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3069194966342527e+26\n"
     ]
    }
   ],
   "source": [
    "R_18=(lambdabar_F*(P**2)*np.e)/8\n",
    "print(R_18)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 233,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.814149979454358"
      ]
     },
     "execution_count": 233,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Radius in Gly\n",
    "R_18/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $4P^3l_{Wi}ƛe/R_N$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0010632074683544304"
      ]
     },
     "execution_count": 234,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# l_wi\n",
    "((2.897771955*10**-3)/2.7255)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 235,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0010630824999999999\n"
     ]
    }
   ],
   "source": [
    "# l_wi=constants.Wien\n",
    "l_wi=1.0630825*10**-3\n",
    "print(l_wi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 236,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3071337184211224e+26\n"
     ]
    }
   ],
   "source": [
    "R_19=(4*P**3*l_wi*lambdabare)/R_N\n",
    "print(R_19)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.816414305528234"
      ]
     },
     "execution_count": 237,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Radius in Gly\n",
    "R_19/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $ƛe.exp((e^{4e-1/a}-ln^2(P^4/a^3))/2)^{1/2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "52356.42609346239"
      ]
     },
     "execution_count": 238,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(e**(4*e-1/a))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 239,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2658205894579623e+83"
      ]
     },
     "execution_count": 239,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P**4/a**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 240,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "36614.930925762965"
      ]
     },
     "execution_count": 240,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.log(P**4/a**3)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3066870223677776e+26\n"
     ]
    }
   ],
   "source": [
    "R_20=np.exp(np.sqrt((e**(4*e-1/a)-np.log(P**4/a**3)**2)/2))*lambdabare\n",
    "print(R_20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3066870223677776e+26\n"
     ]
    }
   ],
   "source": [
    "# R_20=(lambdabare*np.exp((e**(4*e-1/a)-np.log(P**4/a**3)**2))/2)**0.5\n",
    "print(R_20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.811692724518824"
      ]
     },
     "execution_count": 243,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Radius in Gly\n",
    "R_20/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $ƛ_F(Pe^e)^2/√(pn)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 244,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.739167620022749e-19\n"
     ]
    }
   ],
   "source": [
    "lambdabar_f=lambdabare/F\n",
    "print(lambdabar_f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 245,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.939073444929389e+44"
      ]
     },
     "execution_count": 245,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e*(P*e**e)**2/np.sqrt(p*n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 246,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.3067740972914103e+26\n"
     ]
    }
   ],
   "source": [
    "R_21=lambdabar_f*e*(P*e**e)**2/np.sqrt(p*n)\n",
    "print(R_21)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 247,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.812613107187849"
      ]
     },
     "execution_count": 247,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Radius in Gly\n",
    "R_21/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 248,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.1189663868972993e+292"
      ]
     },
     "execution_count": 248,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a**a/np.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Measurement Confirmation of the methods given above by:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Confirmation Measurement #1 (WMAP)\n",
    "https://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe\n",
    "\n",
    "$R_{wmap}=13.74$ Â±0.11 Gly (over 9 years)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Confirmation Measurement #2\n",
    "https://en.wikipedia.org/wiki/Type_Ia_supernova\n",
    "Universe Radius measurement based on supernovae 1a\n",
    "$$R_{SN1a}=13.6(6)Gly$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Confirmation Measurement #3 (Planck Space Observatory)\n",
    "https://en.wikipedia.org/wiki/Planck_(spacecraft)\n",
    "13.8 billion years (as of 2015) – 13.799±0.021 billion (109) years within the Lambda-CDM concordance model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$R_{Planck}=13.799±0.021 Gly$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Confirmation Measurement #4 (James Webb Space Telescope)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### https://en.wikipedia.org/wiki/James_Webb_Space_Telescope"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Measurements due to start in 2021"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Other estimates"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Universe mass calculus given by the relation:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$m_P$ is the Planck Mass , $m_e$ the electron mass, $m_p$ the proton mass, $m_n$ the neutron mass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 249,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{align}\n",
       "\\frac{m_P^4}{m_e\\cdot m_p\\cdot\\ m_n} = M_U \\\\\n",
       "\\end{align}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%latex\n",
    "\\begin{align}\n",
    "\\frac{m_P^4}{m_e\\cdot m_p\\cdot\\ m_n} = M_U \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 250,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.176434e-08\n"
     ]
    }
   ],
   "source": [
    "# Planck's mass\n",
    "m_P=2.176434*10**(-8)\n",
    "print(m_P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 251,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.2437889614753534e-31\n"
     ]
    }
   ],
   "source": [
    "print(m_P**4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 252,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.5520124761382513e-84"
      ]
     },
     "execution_count": 252,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "constants.m_e*constants.m_p*constants.m_n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Universe Mass $8.792233511611562.10^{52}$ kg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 253,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8.792233511611562e+52\n"
     ]
    }
   ],
   "source": [
    "M_U=(m_P**4)/(constants.m_e*constants.m_p*constants.m_n)\n",
    "print(M_U)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## GrandCosmos radius calculus given by the relation:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $R_{GC} = R_N^2/(2.l_P)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 254,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9.075773400070832e+86\n"
     ]
    }
   ],
   "source": [
    "R_GC=R_N**2/(2*l_P)\n",
    "print(R_GC)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 255,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.59310005329329e+61"
      ]
     },
     "execution_count": 255,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_GC/(constants.light_year*10**9)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Volume calculus of R_GC: $V_{GC}=(R_{GC}/r_H)^3(4\\pi/3)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 256,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "137.11\n"
     ]
    }
   ],
   "source": [
    "#aprime=a/(H/p)\n",
    "aprime=137.11\n",
    "print(aprime)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 257,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5.2946297230117714e-11\n"
     ]
    }
   ],
   "source": [
    "# Hydrogen Atom Radius\n",
    "r_H=(lambdabare*aprime)\n",
    "print(r_H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 258,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.109758578455705e+292\n"
     ]
    }
   ],
   "source": [
    "# Volume of R_GC\n",
    "V_GC=((R_GC/r_H)**3)*(4*np.pi/3)\n",
    "print(V_GC)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 259,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.1189663868972993e+292"
      ]
     },
     "execution_count": 259,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a**a/np.pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 260,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{eqnarray}\n",
       "\\ (R_{GC}/r_H)^3(4\\pi/3) \\approx a^a/\\pi \\\\\n",
       "\\end{eqnarray}"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "execution_count": 260,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Latex\n",
    "Latex(r\"\"\"\\begin{eqnarray}\n",
    "\\ (R_{GC}/r_H)^3(4\\pi/3) \\approx a^a/\\pi \\\\\n",
    "\\end{eqnarray}\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 261,
   "metadata": {},
   "outputs": [],
   "source": [
    "aprime=137.1106292"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 262,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1837.15266013165\n"
     ]
    }
   ],
   "source": [
    "H=p+1 -1/(2*(aprime))**2\n",
    "print(H)\n",
    "# p + 1 – 1/2(aH/p)^2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 263,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "292.8061605778735"
      ]
     },
     "execution_count": 263,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Fermi's Energy in GeV\n",
    "np.sqrt(1/(1.1663787*10**-5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 264,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "573007.3624962899"
      ]
     },
     "execution_count": 264,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ratio of Fermi/Electron (electron in MeV)\n",
    "(np.sqrt(1/(1.1663787*10**-5))/0.510998950)*1000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 281,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "List of dividers of 28 :\n",
      "D(28) = [1, 2, 4, 7, 14]\n",
      "phi(1)= 1\n",
      "phi(2)= 1\n",
      "phi(4)= 2\n",
      "phi(7)= 6\n",
      "phi(14)= 6\n",
      "Sum of all phi(d|28) -except 28- = 16\n",
      "                Implying\n",
      "            phi(28) = 12\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/srv/conda/envs/notebook/lib/python3.7/site-packages/ipykernel_launcher.py:24: DeprecationWarning: fractions.gcd() is deprecated. Use math.gcd() instead.\n"
     ]
    }
   ],
   "source": [
    "#!/usr/bin/env python\n",
    "\n",
    "from math import sqrt\n",
    "from fractions import gcd\n",
    "\n",
    "def dividers(n):\n",
    "    D,racine=[1],int(sqrt(n))\n",
    "    fin=racine+1\n",
    "    for d in range(2,fin):\n",
    "        if n%d==0 :\n",
    "            D.extend([d,n//d])\n",
    "    D=sorted(set(D))\n",
    "    print(\"List of dividers of\",n,\":\")\n",
    "    print (\"D(\"+str(n)+\") =\",D)\n",
    "    print\n",
    "    return D\n",
    "\n",
    "def sum_indicatrices(n):\n",
    "    D=dividers(n)\n",
    "    S_phi=0\n",
    "    for d in D:\n",
    "        Nb_inf,p=range(1,d+1),0\n",
    "        for i in Nb_inf:\n",
    "            p+=(gcd(i,d)==1)\n",
    "        print(\"phi(\"+str(i)+\")=\",p)\n",
    "        S_phi+=p\n",
    "    print\n",
    "    print(\"Sum of all phi(d|\"+str(n)+\") -except\", str(n)+\"- =\", S_phi)\n",
    "    print (\"                Implying\")\n",
    "    print (\"            phi(\"+str(n)+\") =\",n-S_phi)\n",
    "    return\n",
    "#n=495\n",
    "#n=33550335\n",
    "n=28 # perfect number 6 28 496 8128 33550336 8589869056\n",
    "\n",
    "sum_indicatrices(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* List of dividers of 8129 :\n",
    "* D(8129) = [1, 11, 739]\n",
    "* phi(1)= 1\n",
    "* phi(11)= 10\n",
    "* phi(739)= 738\n",
    "* Sum of all phi(d|8129) -except 8129- = 749\n",
    "*                Implying\n",
    "*            phi(8129) = 7380"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Sum of all phi(d|33550335) -except 33550335- = 19398015\n",
    "*                Implying\n",
    "*            phi(33550335) = 14152320"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Sum of all phi(d|33550336) -except 33550336- = 16777216\n",
    "*                 Implying\n",
    "*             phi(33550336) = 16773120"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Sum of all phi(d|8128) -except 8128- = 4096\n",
    "*                 Implying\n",
    "*             phi(8128) = 4032"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* List of dividers of 497 :\n",
    "* D(497) = [1, 7, 71]\n",
    "* phi(1)= 1\n",
    "* phi(7)= 6\n",
    "* phi(71)= 70\n",
    "* Sum of all phi(d|497) -except 497- = 77\n",
    "*               Implying\n",
    "*            phi(497) = 420"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Sum of all phi(d|496) -except 496- = 256\n",
    "*                Implying\n",
    "*            phi(496) = 240"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Sum of all phi(d|495) -except 495- = 255\n",
    "*                 Implying\n",
    "*            phi(495) = 240"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the following example, one can write the equation with Latex:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 266,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{eqnarray}\n",
       "\\ R = \\frac{2\\cdot \\hbar^2}{G.m_e.m_p.m_n} \\\\\n",
       "\\ M_U = \\frac{{R}{c}^2}{2G} = \\frac{{m_P}^4}{m_em_pm_n} \\\\\n",
       "\\frac{r_H}{ƛ_e} = \\frac{\\sum\\frac{1}{n}}{\\sum\\frac{1}{n^2}}\\\\\n",
       "\\end{eqnarray}"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "execution_count": 266,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Latex\n",
    "Latex(r\"\"\"\\begin{eqnarray}\n",
    "\\ R = \\frac{2\\cdot \\hbar^2}{G.m_e.m_p.m_n} \\\\\n",
    "\\ M_U = \\frac{{R}{c}^2}{2G} = \\frac{{m_P}^4}{m_em_pm_n} \\\\\n",
    "\\frac{r_H}{ƛ_e} = \\frac{\\sum\\frac{1}{n}}{\\sum\\frac{1}{n^2}}\\\\\n",
    "\\end{eqnarray}\"\"\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    ":star:\n",
    "\n",
    "# Docs\n",
    "\n",
    "This project aims at providing a repository dedicated to Dr Francis Michel Sanchez's work in holography and cosmology; it is just a mere continuation of what was initiated and published in a set of theoritical physics articles.\n",
    "\n",
    "\n",
    "## Publications\n",
    "* Please, cite this paper (http://vixra.org/pdf/1811.0146v8.pdf) if you employ this package in your research work.\n",
    "* [Vixra Publications](http://vixra.org/author/francis_m_sanchez)\n",
    "* [Rxiv Publications](http://rxiv.org/author/f_m_sanchez)\n",
    "* [Nature: Observations of solar oscillations with periods of 160 minutes](https://www.nature.com/articles/277635a0)\n",
    "* [A coherent resonant cosmology approach and its implications in microphysics and Biophysics Pages 375-407 of Quantum Systems in Physics, Chemistry](https://www.springer.com/fr/book/9783319502540?token=gbgen&wt_mc=GoogleBooks.GoogleBooks.3.EN#)\n",
    "* [From Coherent Cosmic Oscillations to the Steady-State Cyclic Ultrafast](https://rxiv.org/abs/1401.0228)\n",
    "* [Evidence for a Steady-State, Holographic, Tachyonic and Super-Symmetric Cosmology](https://www.researchgate.net/publication/267430152_Evidence_for_a_Steady-State_Holographic_Tachyonic_and_Super-Symmetric_Cosmology)\n",
    "* [academia.edu Correcting Five Historical Faults in Modern Cosmology](https://www.academia.edu/30843365/Correcting_Five_Historical_Faults_in_Modern_Cosmology)\n",
    "\n",
    "## Videos\n",
    "\n",
    "* [L'enigme des grands nombres Part1](https://www.youtube.com/watch?v=ev_eIqLtK8s)\n",
    "* [L'enigme des grands nombres Part2](https://www.youtube.com/watch?v=i4w392nYqns)\n",
    "* [L'enigme des grands nombres Part3](https://www.youtube.com/watch?v=b4WvI0yVXnE)\n",
    "* [Exposés de Francis Sanchez playlist created by J.Bailhache](https://www.youtube.com/playlist?list=PLQXUNrzoAeFbmfHo3jgWzzExlVoBn2JUw)\n",
    "* [Khelif/Torgoz/Sanchez' Debate on LE FORMALISME EN PHYSIQUE ET EN SCIENCE](https://www.youtube.com/watch?v=TEK2hCI2we0)\n",
    "\n",
    "* [La science du 3eme millenaire playlist created by J.Bailhache](https://www.youtube.com/playlist?list=PLQXUNrzoAeFbLOuEsltAUWJWCcAEgeGJE)\n",
    "* [Francis Sanchez Cosmologie 23/03/2017](https://www.youtube.com/watch?v=NmvvIQjxBw8)\n",
    "* [La complexité mathématique face à la simplicité cosmique - Exposé de Francis Sanchez](https://www.youtube.com/watch?v=dyUlUahgHGg)\n",
    "* [La méthode scientifique en question et les constantes fondamentales - Exposé](https://www.youtube.com/watch?v=zUt1hxTdiuA)\n",
    "* [Jean Maruani chez lui et Francis Sanchez autour du nombre 237 29. 12. 17](https://www.youtube.com/watch?v=05g416VWjnw)\n",
    "* [Pr. Michel Grosmann: Holographie quanto-cosmique](https://www.youtube.com/watch?v=M7bJnekDIt0)\n",
    "* [Pr. Renée Veysseyre: La constante d'Atiyah réunifie Physique et Mathématique](https://www.youtube.com/watch?v=mbPpKqakDAU)\n",
    "* [Christian Bizouard, astronome: Omniprésence astrophysique du cycle de Kotov](https://www.youtube.com/watch?v=DGeBC73kKvQ)\n",
    "* [Pr. Francis Sanchez: Historique, Principe Calculatoire, Axe Topologique et Principe Holique](https://www.youtube.com/watch?v=IgKSrfsqFEs)\n",
    "\n",
    "## Books\n",
    "\n",
    "* [Bizouard, Kotov, Sanchez wrote in 20th Natural Philosophy Alliance Proceedings by David de Hilster, published by Lulu.com in 2013]()\n",
    "* [Quantum Systems in Physics, Chemistry, Springer Ed](https://www.amazon.com/gp/search?index=books&linkCode=qs&keywords=9783319502557)\n",
    "* [The Worldwide List of Alternative Theories and Critics](https://books.google.com/books?isbn=2902425171)\n",
    "* [The Dirac Electron: From Quantum Chemistry to Holistic Cosmology by J.Maruani](https://onlinelibrary.wiley.com/doi/abs/10.1002/jccs.201500374#)\n",
    "* [solar 22 years cycle](https://link.springer.com/article/10.1007%2Fs10509-016-2985-8)\n",
    "* [Current Issues in Cosmology](https://www.cambridge.org/gb/academic/subjects/physics/cosmology-relativity-and-gravitation/current-issues-cosmology?format=HB&isbn=9780521858984)\n",
    "* ```@inproceedings{Sanchez2015RemarkablePO,\n",
    "  title=Remarkable Properties of the Eddington Number 137 and Electric Parameter 137 . 036 excluding the Multiverse Hypothesis},\n",
    "  author={Francis M. Sanchez and Valery A. Kotov,\n",
    "  year=2015```\n",
    "\n",
    "## Papers\n",
    "\n",
    "* [College de France (2004): Discussions sur les oscillations cosmiques, les nombres sans dimension et les periodicites en microphysique et cosmologie - College de France 27th Feb 2004](https://lempel.pagesperso-orange.fr/occ_cdf.pdf)\n",
    "\n",
    "More to come soon stay tuned.../.:stars:."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "========================================================================================================================="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 267,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{eqnarray}\n",
       "\\frac{m_p^4}{m_p\\cdot m_H\\cdot\\ m_e} = M \\\\\n",
       "\\frac{\\pi_Pt}{\\pi_Bb} = 6 \\cdot \\frac{\\pi^5}{p_H^9} \\\\\n",
       "\\end{eqnarray}"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "execution_count": 267,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Write your equation in a dedicated cell with latex\n",
    "\n",
    "from IPython.display import Latex\n",
    "Latex(r\"\"\"\\begin{eqnarray}\n",
    "\\frac{m_p^4}{m_p\\cdot m_H\\cdot\\ m_e} = M \\\\\n",
    "\\frac{\\pi_Pt}{\\pi_Bb} = 6 \\cdot \\frac{\\pi^5}{p_H^9} \\\\\n",
    "\\end{eqnarray}\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 268,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{eqnarray}\n",
       "S = \\frac{c^3 k A}{4 \\hbar G} \\\\\n",
       "\\end{eqnarray}"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "execution_count": 268,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Write Bekenstein-Hawking Entropy \n",
    "from IPython.display import Latex\n",
    "Latex(r\"\"\"\\begin{eqnarray}\n",
    "S = \\frac{c^3 k A}{4 \\hbar G} \\\\\n",
    "\\end{eqnarray}\"\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}