Fie \( p\geq 2 \) un numar natural. Sa se arate ca \( p \) este prim daca si numai daca orice poligon echiangular cu lungimile laturilor numere rationale cu \( p \) laturi este regulat.
Mihai Piticari
Alta definitie a unui numar prim
-
Cezar Lupu
- Site Admin
- Posts: 612
- Joined: Wed Sep 26, 2007 2:04 pm
- Location: Bucuresti sau Constanta
- Contact:
Alta definitie a unui numar prim
Last edited by Cezar Lupu on Tue Mar 25, 2008 5:32 pm, edited 1 time in total.
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
-
Beniamin Bogosel
- Co-admin
- Posts: 710
- Joined: Fri Mar 07, 2008 12:01 am
- Location: Timisoara sau Sofronea (Arad)
- Contact:
Consideram \( \varepsilon=\cos \frac{2\pi}{p}+i\sin \frac{2 \pi}{p} \). Conditia ca sa existe un poligon echiangular cu \( p \) laturi numere rationale (cel putin doua distincte) este echivalenta cu existenta numerelor rationale distincte \( q_i,\ i=0..p-1 \) astfel incat \( \sum_{i=0}^{p-1} \varepsilon^i q_i =0 \). Acest lucru este echivalent cu faptul ca polinomul minimal al lui \( \varepsilon \) peste \( \mathbb{Q} \) nu este \( 1+X+...+X^{p-1} \). Acest lucru este echivalent cu faptul ca \( p \) nu este prim.
Negand aceasta echivalenta, obtinem chiar problema noastra.
Negand aceasta echivalenta, obtinem chiar problema noastra.
Yesterday is history,
Tomorow is a mistery,
But today is a gift.
That's why it's called present.
Blog
Tomorow is a mistery,
But today is a gift.
That's why it's called present.
Blog
-
Bogdan Cebere
- Thales
- Posts: 145
- Joined: Sun Nov 04, 2007 1:04 pm
Despre aceasta problema si alte aplicatii ale poligoanelor echiunghiulare se poate citi in GMB nr.11/2002, in articolul semnat de Titu Andreescu si Bogdan Enescu.
Last edited by Bogdan Cebere on Wed May 20, 2009 4:11 pm, edited 1 time in total.