Solution pour le challenge 79
On peut décomposer tout entier en produit de facteurs premiers :
![Rendered by QuickLaTeX.com p_{1},\cdots,p_{r}](https://math-os.com/wp-content/ql-cache/quicklatex.com-b45d951a29b2b491b5361789b2364752_l3.png)
![Rendered by QuickLaTeX.com \alpha_{1},\cdots,\alpha_{r}\in\mathbb{N}^{\star}.](https://math-os.com/wp-content/ql-cache/quicklatex.com-ee23c78ea449154e296fb4d891bffcb7_l3.png)
![Rendered by QuickLaTeX.com n](https://math-os.com/wp-content/ql-cache/quicklatex.com-a44d662e2fcd865f31268b1147c8a4be_l3.png)
Comme est un nombre premier, la condition
impose
et
Ainsi
est nécessairement de la forme :
![Rendered by QuickLaTeX.com p](https://math-os.com/wp-content/ql-cache/quicklatex.com-37f6a1a9a39aa263d588cacc9c82b405_l3.png)
![Rendered by QuickLaTeX.com p=2.](https://math-os.com/wp-content/ql-cache/quicklatex.com-5be7f69849d71788d8f45f69ced96a95_l3.png)
![Rendered by QuickLaTeX.com 61](https://math-os.com/wp-content/ql-cache/quicklatex.com-2b76abd4decf457b9d490fa16b33e5ea_l3.png)
Plus généralement, si est premier, alors le plus petit entier positif possédant
diviseurs est
On peut encore généraliser …
Notons désormais le
ème nombre premier (de sorte que
etc …). Soient
des nombres premiers distincts et soit
leur produit (par exemple :
Si
vérifie
c’est-à-dire :
![Rendered by QuickLaTeX.com r=s](https://math-os.com/wp-content/ql-cache/quicklatex.com-2cea6246f87624410d0568646caa560a_l3.png)
![Rendered by QuickLaTeX.com 1+\alpha_{i}](https://math-os.com/wp-content/ql-cache/quicklatex.com-b287136bab4186e0167af8a98c2219c9_l3.png)
![Rendered by QuickLaTeX.com 1\leqslant i\leqslant s)](https://math-os.com/wp-content/ql-cache/quicklatex.com-8a05036ff6997b601b0e973a63aa25f3_l3.png)
![Rendered by QuickLaTeX.com q_{i},](https://math-os.com/wp-content/ql-cache/quicklatex.com-a3e98b5f996abe877d4b29bccf5262c3_l3.png)
![Rendered by QuickLaTeX.com n](https://math-os.com/wp-content/ql-cache/quicklatex.com-a44d662e2fcd865f31268b1147c8a4be_l3.png)
()
![Rendered by QuickLaTeX.com q_{1}>q_{2}>\cdots>q_{s}](https://math-os.com/wp-content/ql-cache/quicklatex.com-0f94c4ae1ec742c2c06bff7cc65a968c_l3.png)
![Rendered by QuickLaTeX.com \left(\star\right)](https://math-os.com/wp-content/ql-cache/quicklatex.com-d928fdc3003a591d91c4792b57b630c9_l3.png)
![Rendered by QuickLaTeX.com p_{i}=\pi_{i}](https://math-os.com/wp-content/ql-cache/quicklatex.com-c2b3764c5ace0dbba75ad4877a3fa995_l3.png)
![Rendered by QuickLaTeX.com i\in\left\llbracket 1,s\right\rrbracket .](https://math-os.com/wp-content/ql-cache/quicklatex.com-32ce56cca92def8c152fad2660530d40_l3.png)
Pour consulter l’énoncé, c’est ici