Lhermite's models

Lhermite's models are interesting ways to synthesize various objects that are apparently scattered.

Prime numbers and the model of three arrows

${𝕌}_n=\sum _{i=1}^{f\left(n\right)}(\left[\frac{1+\sum _{m=1}^i\phi \left(m\right)}{n+1}\right]\times \left[\frac{n+1}{1+\sum _{m=1}^i\phi \left(m\right)}\right]\times i\times \phi \left(i\right))$

${𝕌}_n=\sum _{i=1}^{f\left(n\right)}(\left[\frac{\alpha +\sum _{m=1}^i\phi \left(m\right)}{n+\alpha }\right]\times \left[\frac{n+\alpha }{\alpha +\sum _{m=1}^i\phi \left(m\right)}\right]\times i\times \phi \left(i\right))$

with $f\left(n\right)\ge {𝕌}_n$ and $\alpha >0$

$P_n=\sum _{i=1}^{2^{2^n}}(\lfloor \frac{1+\sum _{m=1}^i(1-\lfloor \frac{\lfloor \frac{{(m!)}^2}{m^3}\rfloor }{\frac{{(m!)}^2}{m^3}}\rfloor )}{n+1}\rfloor \times \lfloor \frac{n+1}{1+\sum _{m=1}^i(1-\lfloor \frac{\lfloor \frac{{(m!)}^2}{m^3}\rfloor }{\frac{{(m!)}^2}{m^3}}\rfloor )}\rfloor \times i\times \left(1-\lfloor \frac{\lfloor (\frac{{(i!)}^2}{i^3}\rfloor }{\frac{{(i!)}^2}{i^3}}\rfloor \right))$

$P_n=\sum _{i=1}^{2^n}(\lfloor \frac{1+\sum _{m=1}^i(1-\lfloor \frac{\lfloor \frac{{(m!)}^2}{m^3}\rfloor }{\frac{{(m!)}^2}{m^3}}\rfloor )}{n+1}\rfloor \times \lfloor \frac{n+1}{1+\sum _{m=1}^i(1-\lfloor \frac{\lfloor \frac{{(m!)}^2}{m^3}\rfloor }{\frac{{(m!)}^2}{m^3}}\rfloor )}\rfloor \times i\times \left(1-\lfloor \frac{\lfloor (\frac{{(i!)}^2}{i^3}\rfloor }{\frac{{(i!)}^2}{i^3}}\rfloor \right))$

$P_n=\sum _{i=1}^{1+n!}(\lfloor \frac{1+\sum _{m=1}^i(1-\lfloor \frac{\lfloor \frac{{(m!)}^2}{m^3}\rfloor }{\frac{{(m!)}^2}{m^3}}\rfloor )}{n+1}\rfloor \times \lfloor \frac{n+1}{1+\sum _{m=1}^i(1-\lfloor \frac{\lfloor \frac{{(m!)}^2}{m^3}\rfloor }{\frac{{(m!)}^2}{m^3}}\rfloor )}\rfloor \times i\times \left(1-\lfloor \frac{\lfloor (\frac{{(i!)}^2}{i^3}\rfloor }{\frac{{(i!)}^2}{i^3}}\rfloor \right))$

$P_n=\sum _{i=1}^{2^n}(\left[\frac{1+\sum _{m=1}^i(1-\left[\frac{\left[\frac{{(m!)}^2}{m^3}\right]}{\frac{{(m!)}^2}{m^3}}\right])}{n+1}\right]\times \left[\frac{n+1}{1+\sum _{m=1}^i(1-\left[\frac{\left[\frac{{(m!)}^2}{m^3}\right]}{\frac{{(m!)}^2}{m^3}}\right])}\right]\times i\times \left(1-\left[\frac{[(\frac{{(i!)}^2}{i^3}]}{\frac{{(i!)}^2}{i^3}}\right]\right))$

$P_n=\sum _{i=1}^{2^{2^n}}(\left[\frac{1+\sum _{m=1}^i(1-\left[\frac{\left[\frac{{(m!)}^2}{m^3}\right]}{\frac{{(m!)}^2}{m^3}}\right])}{n+1}\right]\times \left[\frac{n+1}{1+\sum _{m=1}^i(1-\left[\frac{\left[\frac{{(m!)}^2}{m^3}\right]}{\frac{{(m!)}^2}{m^3}}\right])}\right]\times i\times \left(1-\left[\frac{[(\frac{{(i!)}^2}{i^3}]}{\frac{{(i!)}^2}{i^3}}\right]\right))$

Red balls and blue balls and prime numbers

$P_{((1-\left[\frac{\left[\frac{{(n!)}^2}{n^3}\right]}{\frac{{(n!)}^2}{n^3}}\right])\times (\sum _{m=1}^n(1-\left[\frac{\left[\frac{{(m!)}^2}{m^3}\right]}{\frac{{(m!)}^2}{m^3}}\right])-i)+i)}=(P_i-n)\times \left[\frac{\left[\frac{{(n!)}^2}{n^3}\right]}{\frac{{(n!)}^2}{n^3}}\right]+n$

Prime numbers and the model of three arrows according to Wilson's theorem

$\mathrm{\forall }n\in {\mathbb{N}}^{\prime }$

$(n-1)!\equiv -1(\mathrm{mod}n)\iff n\in \mathbb{P}$

in the same way, it is advanced that

$\mathrm{\forall }n\in {\mathbb{N}}^{\prime }$

$\left[\frac{\left[\frac{(n-1)!+1}{n}\right]}{\frac{(n-1)!+1}{n}}\right]=1\iff n\in \mathbb{P}$

It's very evident that

$\mathrm{\forall }n\in {\mathbb{N}}^{\prime }$

$\left[\frac{\left[\frac{(n-1)!+1}{n}\right]}{\frac{(n-1)!+1}{n}}\right]=0\iff n\notin \mathbb{P}$

Therefore, according to Lhermite's models and Wilson's theorem, there are two evident theorems :

$\mathrm{\forall }n\in {\mathbb{N}}^{\mathbb{*}}$

$\left[\frac{\left[\frac{(n-1)!+1}{n}\right]}{\frac{(n-1)!+1}{n}}\right]-\left[\frac{1}{n}\right]=1\iff n\in \mathbb{P}$

$\mathrm{\forall }n\in {\mathbb{N}}^{\mathbb{*}}$

$\left[\frac{\left[\frac{(n-1)!+1}{n}\right]}{\frac{(n-1)!+1}{n}}\right]-\left[\frac{1}{n}\right]=0\iff n\notin \mathbb{P}$

Therefore the following relation becomes true :

$\mathrm{\forall }n\in {\mathbb{N}}^{\mathbb{*}}$

$\left[\frac{\left[\frac{(n-1)!+1}{n}\right]}{\frac{(n-1)!+1}{n}}\right]-\left[\frac{1}{n}\right]=1-\left[\frac{\left[\frac{{(n!)}^2}{n^3}\right]}{\frac{{(n!)}^2}{n^3}}\right]$

Let's choose one of the formulas that are indicated in the first section :

$P_n=\sum _{i=1}^{2^{2^n}}(\left[\frac{1+\sum _{m=1}^i(1-\left[\frac{\left[\frac{{(m!)}^2}{m^3}\right]}{\frac{{(m!)}^2}{m^3}}\right])}{n+1}\right]\times \left[\frac{n+1}{1+\sum _{m=1}^i(1-\left[\frac{\left[\frac{{(m!)}^2}{m^3}\right]}{\frac{{(m!)}^2}{m^3}}\right])}\right]\times i\times \left(1-\left[\frac{[(\frac{{(i!)}^2}{i^3}]}{\frac{{(i!)}^2}{i^3}}\right]\right))$

let's replace

$1-\left[\frac{\left[\frac{{(m!)}^2}{m^3}\right]}{\frac{{(m!)}^2}{m^3}}\right]by\left[\frac{\left[\frac{(m-1)!+1}{m}\right]}{\frac{(m-1)!+1}{m}}\right]-\left[\frac{1}{m}\right]$

and

$1-\left[\frac{\left[\frac{{(i!)}^2}{i^3}\right]}{\frac{{(i!)}^2}{i^3}}\right]by\left[\frac{\left[\frac{(i-1)!+1}{i}\right]}{\frac{(i-1)!+1}{i}}\right]-\left[\frac{1}{i}\right]$

Therefore an equivalent expression is :

$P_n=\sum _{i=1}^{2^{2^n}}(\left[\frac{1+\sum _{m=1}^i(\left[\frac{\left[\frac{(m-1)!+1}{m}\right]}{\frac{(m-1)!+1}{m}}\right]-\left[\frac{1}{m}\right])}{n+1}\right]\times \left[\frac{n+1}{1+\sum _{m=1}^i(\left[\frac{\left[\frac{(m-1)!+1}{m}\right]}{\frac{(m-1)!+1}{m}}\right]-\left[\frac{1}{m}\right])}\right]\times i\times (\left[\frac{\left[\frac{(i-1)!+1}{i}\right]}{\frac{(i-1)!+1}{i}}\right]-\left[\frac{1}{i}\right]))$

Function Ω according to Lhermite's models

$\mathrm{\Omega }\left(n\right)=\sum _{j=1}^n\left(\sum _{i=1}^n\left(\left[\frac{\left[\frac{n}{i^j}\right]}{\left(\frac{n}{i^j}\right)}\right]\times (1-\left[\frac{\left[\frac{{(i!)}^2}{i^3}\right]}{\frac{{(i!)}^2}{i^3}}\right])\right)\right)$

Liouville's function and Lhermite's models

$\lambda \left(n\right)={(-1)}^{\left(\sum _{j=1}^n\left(\sum _{i=1}^n\left(\left[\frac{\left[\frac{n}{i^j}\right]}{\left(\frac{n}{i^j}\right)}\right]\times (1-\left[\frac{\left[\frac{{(i!)}^2}{i^3}\right]}{\frac{{(i!)}^2}{i^3}}\right])\right)\right)\right)}$

Three Arrows or Jonatan's Arrows

There are three possibilities :$a>b$ or $b<a$ or $a=b$ . In the same way , there are three possibilities : $V_i>n$ or $V_i<n$ or $V_i=n$

with

See also

• http://ht.wikipedia.org/wiki/Fonksyon_ki_konte_kantite_divizè_n

• http://ht.wikipedia.org/wiki/Fonksyon_ki_konte_kantite_nonm_premye_jimo_yo

• http://ht.wikipedia.org/wiki/_Fonksyon_ki_konte_nonm_premye_ki_sou_fòm_an+b_lè_a_ak_b_premye_entre_yo

• http://ht.wikipedia.org/wiki/Fonksyon_lanmbda_Liouville

• http://ht.wikipedia.org/wiki/Fonksyon_

• http://ht.wikipedia.org/wiki/Fonksyon_Omega_ki_konte_kantite_faktè_premye_ki_rantre_nan_dekonposition_yon_nonm_antye_n

• http://ht.wikipedia.org/wiki/Fonksyon_omega_ki_konte_kantite_nonm_premye_ki_diferan_ki_divize_yon_nonm_n

• http://ht.wikipedia.org/wiki/Fonksyon_phi_Euler

• http://ht.wikipedia.org/wiki/Fonksyon_pi,_ki_konte_kantite_nonm_premye

• http://ht.wikipedia.org/wiki/Fonksyon_psi_Tchebycheff

• http://ht.wikipedia.org/wiki/Fonksyon_tèta_Tchebycheff

• http://ht.wikipedia.org/wiki/Fonksyon_Von_Mangolt

• http://ht.wikipedia.org/wiki/Fòmil_pou_nonm_konpoze

• http://ht.wikipedia.org/wiki/Modèl_boul_wouj_ak_boul_ble

• http://ht.wikipedia.org/wiki/Modèl_flèch_Jonatan

• http://ht.wikipedia.org/wiki/Modèl_Lhermite_yo

• http://ht.wikipedia.org/wiki/Modèl_mouton_lòt_patiraj

• http://ht.wikipedia.org/wiki/Modèl_pitit_prodig_ou_modèl_ti_mouton_pèdi

• http://ht.wikipedia.org/wiki/Modèl_predesesè

• http://ht.wikipedia.org/wiki/Modèl_siksesè

• http://ht.wikipedia.org/wiki/Modèl_ti_triyang_ak_gwo_triyang

• http://ht.wikipedia.org/wiki/Modèl_triyang_ak_kwadrilatè

• http://ht.wikipedia.org/wiki/Nonm_pafè

• http://ht.wikipedia.org/wiki/Nonm_premye

• http://ht.wikipedia.org/wiki/Nonm_premye_ak_modèl_boul_rouj_ak_boul_ble

• http://ht.wikipedia.org/wiki/Nonm_premye_Ezau

• http://ht.wikipedia.org/wiki/Nonm_premye_Fermat

• http://ht.wikipedia.org/wiki/Nonm_premye_Izaak

• http://ht.wikipedia.org/wiki/Nonm_premye_Jakòb

• http://ht.wikipedia.org/wiki/Nonm_premye_Mersenne

• http://ht.wikipedia.org/wiki/Nonm_zanmi

• http://ht.wikipedia.org/wiki/Primoryel

• http://ht.wikipedia.org/wiki/Pwisansyèl

• http://ht.wikipedia.org/wiki/Rezolisyon_ekasyon_premye_degre_ak_sentèz

• http://ht.wikipedia.org/wiki/Rezolisyon

• http://ht.wikipedia.org/wiki/Fonksyon_ki_konte_nonm_premye_ki_sou_fòm_an+b_lè_a_ak_b_premye_entre_yo