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Numere prime

 
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Marius Mainea
Gauss


Joined: 26 May 2008
Posts: 1099
Location: Gaesti (Dambovita)

PostPosted: Sat Dec 27, 2008 1:08 pm    Post subject: Numere prime Reply with quote

Sa se arate ca daca p si q sunt doua numere naturale prime atunci numarul p^{2q}+q^{2p} nu este prim.
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mumble
Euclid


Joined: 30 Jan 2008
Posts: 48

PostPosted: Wed Jun 24, 2009 7:34 pm    Post subject: Reply with quote

Vad ca asta a ramas (surprinzator Confused ) cam mult timp nerezolvata.

Daca p^{2q}+q^{2p} ar fi prim cu necesitate unul din numerele p,q, fie el p trebuie sa fie 2, in timp ce q trebuie sa fie impar.
Dar 4^q+q^4=4\cdot 2^{2(q-1)}+q^4. Cum q este impar, ultima expresie este de forma a^4+4b^4 care se descompune conform identitatii Sophie-Germain in (a^2+2b^2-2ab)(a^2+2b^2+2ab), evident un numar compus.
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