Diskite:Modèl Lhermite yo

message: Diskite atik sa a an anglè e ede amelyore
Lhermite's models are interesting ways to synthesize some objects that are apparently scattered.

message: : 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) 
replies: text: 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)

text: 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)

text: 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)

text: 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)
thread_title: Prime numbers and the model of three arrows

message: 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
thread_title: Red balls and blue balls and prime numbers

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

message: : \forall n \in \mathbb{N'} 
replies: text: (n-1)! \equiv\ -1 \pmod n \Leftrightarrow n \in \mathbb{P}  
in the same way, it is advanced that

text: \forall n \in \mathbb{N'}

text: \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

text: \forall n \in \mathbb{N'}

text: \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 :

text: \forall n \in \mathbb{N^*}

text: \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}

text: \forall n \in \mathbb{N^*}

text: \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 :

text: \forall n \in \mathbb{N^*}

text: \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 :

text: 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
 
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
 
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 :

text: 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) }
thread_title: Prime numbers and the model of three arrows according to Wilson's theorem

message: : \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)
thread_title:  Function &Omega; according to Lhermite's models

message: : \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)}
thread_title: The Liouville function and Lhermite's models

message: Let's explain Jonatan's arrows for any context.
This model has a certain relation with Boole's works and Leibniz work's.
Let's start with the explaination now.
thread_title: Three Arrows or Jonatan's Arrows 

thread_title: References

message: * Bertrand's postulate
* Boole
* Boolean algebra
* Euclid's theorem
* Wilson's theorem
Kategori:Formal sciences
Kategori:Matematik‏‎
thread_title:  See also