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

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

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

with ${\displaystyle f\left(n\right)\geq \mathbb {U} _{n}}$ and ${\displaystyle \alpha >0}$

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

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

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

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

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

Red balls and blue balls and prime numbers

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

Red balls and blue balls and prime numbers according to Wilson's theorem

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

${\displaystyle \forall n\in \mathbb {N'} }$

${\displaystyle (n-1)!\equiv \ -1{\pmod {n}}\Leftrightarrow n\in \mathbb {P} }$

in the same way, it is advanced that

${\displaystyle \forall n\in \mathbb {N'} }$

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

It's very evident that

${\displaystyle \forall n\in \mathbb {N'} }$

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

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

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

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

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

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

Therefore the following relation becomes true :

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

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

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

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

let's replace

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

and

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

Therefore an equivalent expression is :

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

Function Ω according to Lhermite's models

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

Liouville's function and Lhermite's models

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

Three Arrows or Jonatan's Arrows

There are three possibilities :${\displaystyle a>b}$ or ${\displaystyle b<a}$ or ${\displaystyle a=b}$ . In the same way , there are three possibilities : ${\displaystyle V_{i}>n}$ or ${\displaystyle V_{i}<n}$ or ${\displaystyle V_{i}=n}$

with

References

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