name

symbol

value

remarks

Euler-Napier constant and associated optimal numbers, e₂ = $e^e$ ; e₃ = $e^{e^e}$ ; e₄ = $e^{e^{e^e}}$

e

2.718281828459042

$e_4^{1/a{a}^{\prime }}\approx {\left(9/2\right)}^{1/a^{\prime }}\approx P⁴{\sin }^2\theta$

Archimedes constant $\pi$ and ${\pi }_a\approx 6/\left(1/g_1-1\right)\approx 3/\left(1-g_1{g}_2{g}_3/6\right)\approx 3+{\left(q/\left(1+1/{\Delta }_{Ed}\right)\right)}^2/g_2$

$\pi$

3.14159265358979

${\pi }_a\approx {\left(a^2-{137}^2\right)}^{1/2}$

Euler-Mascheroni constant $\gamma /\pi =\Gamma /a\approx \ln \Phi /{\Phi }^2\approx \ln \zeta \left(3\right)$

$\gamma$

0.57721566490153

${\Phi }^{\pi /\gamma }\approx e^{{\Phi }^2}\approx a/10$

Golden ratio and Black Body Photon Constant $\varphi =\lambda /l_{ph}={\left(16\pi \zeta \left(3\right)\right)}^{1/3}$

$\Phi$

1.61803398874990

$\zeta \left(3\right)\approx 1.20205691$

Wien parameter $W_5=hc/k_{B}T{l}_{Wien}=5\left(1-e^{-W_5}\right)\approx a/137g_1^3\approx \ln 2-4\ln g_1$

$w_5$

4.96514245

$W_3\approx 2.821439372$

Relative Radiation Ratio $u_{CMB+CNB}/u_{CMB}=1+3\left(7/8\right){\left(4/11\right)}^{4/3}\approx p_t{n}_t/H{p}_{W1}$

y

1.681321953

$p_{W1}=1903$ (A001220)

Scale-factor $8{\pi }^2/\ln 2\approx {\left(16q\right)}^3\approx i^{\pi /i}/g_3\approx \left(i^{\pi /i}-1\right)/4q\approx \pi {\Delta }_{Ed}/8\approx {\left({\Phi }^4/g_2\right)}^2$

j

113.9106346

[5] $j\approx T_{mam}/T_{CMB}$

Lucas Large Prime Number $P^{3+7}\approx a^7{\left(\sqrt{a}\right)}^{7+127}\Rightarrow P/aw\sqrt{a}\approx {\left({\left(\sqrt{a}\right)}^{127}/P^3\right)}^{1/7}\approx n^3$

$n_L$

2¹²⁷ - 1

[7] Cb. Hier. 3,7,127,2¹²⁷ - 1

Eddington Large Number $\approx {\left(1/g_1-1\right)}^{{\tau }^2/{\mu }^2}1839/2d_e{n}_t\approx {\left(6/\pi \right)}^{{\tau }^2/{\mu }^2-1}{\left(4\pi \right)}^2\sqrt{a}/1839$

$n_{Ed}$

136 × 2²⁵⁶

[10]

Eddington Proton-Electron mass ratio $p_{Ed}=\left(136+{\Delta }_{Ed}\right)/\left(136-{\Delta }_{ed}\right)$ ${\Delta }_{ed}={\left({136}^2-40\right)}^{1/2}$

$p_{Ed}$

1847.599459

[10] ${\Delta }_{ed}={\left({137}^2-40\right)}^{1/2}$

Maximal known Euler idoneal (“Suitable”) Number $\approx p_{Ed}{H}^5/n_t^2{p}_{00}^3\approx {\left(H\Phi \right)}^3/a^{7/2}\sin \theta$

$s_{65}$

1848

OEIS (A000926)

Extended Maximal Number of Intersection of Diagonals from a 16-gone $1836\approx {\left(3a^3/20q\right)}^{1/2}/{\left(g_2{g}_3\right)}^2$

$p_{00}$

1836

OEIS (A000332+n)

Proton/Electron mass ratio $m_p/m_e$ $n_t/p_t\approx {\left(n_t-p_t\right)}^{1/{\left(26p_t/n_t\right)}^2}\approx {\left({\left(4\pi p_W/p_t\right)}^2\right)}^{1/\sqrt{p_t{n}_t}}$

$p_t$

1836.15267343

0.06 ppb

Wyler Proton/Electron mass ratio $p_W=6{\pi }^5\approx {\left(a{p}_t/137\right)}^2/H\approx \sqrt{p_t}{\left(s_{65}\sqrt{p_{00}}/p_{Ed}\right)}^3/H\approx {\left(H{p}_{00}^7\right)}^{1/2}/p_t^3$

$p_W$

1836.118109

$P/a_w\sqrt{a}\approx n_t^3{\left(p_W/p_t\right)}^5$

Hydrogen/Electron mass ratio $H\approx 8e^{2e}\approx 495/g_1{g}_2{g}_3$

H

1837.15266014

0.06 ppb

Relativistic correction factor $1/\left(H-p_t\right)$

$\beta$

1.000026597

0.1 ppb

Neutron/Electron mass ratio $n_t\approx {\left({\pi }^2{W}_5/4\right)}^3\approx p_t+1+1/g_{t0}\approx p_t{\left(61/60\right)}^{1/12}\approx \sqrt{\left(p_{t}H\right)}{d}_e/{\beta }^2$

$n_t$

1838.6836617

0.5 ppb

Planck ratio $m_P/m_e=P\approx 2^{64}{\left(a/4\pi \right)}^3{\beta }^2{p}_t/n_t\approx {\left(\sin \theta \right)}^{-a{d}_e/2}$

P

2.2389015907 × 10²²

ppb [5]

Lucas Gravitational Proton/Electron mass ratio $P/\sqrt{n_L}\approx \sqrt{2}{\left(a/137\right)}^2{p}_t/p_W{q}^6$

$p_G$

1831.531181

$\approx \beta u_{30}a/137$

Bisection of Rule 23 Wolfram series (R23W) $u_d=1+d\left(2d+1\right)$ for d = 30

$u_{30}$

1831

EOIS(A266438)

Combined R23W: $u^{\text{'}}{}_d=u_{2D=2\left(d+3\right)}=1+2\left(d+3\right)+8{\left(d+3\right)}^2$ for d = 12

$u^{\text{'}}{}_{12}$

1831

30D to 12D

R23W for d = 8: Eddington’s formula [10] : $u_8=136+1$

$u_8$

137

$v_8=495+496$

Combined R23W $u^{\text{'}}{}_1\mathrm{<mo>\; =\; </mo>}{u}_{2\times \left(1+3\right)}$ for d = 1: Hol Sweeping 1D Absolute Space

$u^{\text{'}}{}_1$

137

12D to 1D

$u_{2d}=1+2d\left(4d+1\right)$ Bisection of odd central polygonal numbers for d = 4

$a_{00}$

137

OEIS (A188135)

Eddington-Atiyah constant $2^7+2^3+2^0$

$a_{00}$

137

from three algebra [30]

Maximal number of zones defined by 16 straight lines in a plane

$a_{00}$

137

OEIS (AA000124)

Number of partitions of 20 that do not contain 1 as a part

$a_{00}$

137

OEIS (A002865)