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Ecuatie in m si n

 
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BogdanCNFB
Thales


Joined: 07 May 2008
Posts: 141
Location: Craiova

PostPosted: Wed Jan 30, 2013 7:01 pm    Post subject: Ecuatie in m si n Reply with quote

Gasiti m si n naturale astfel incat:
m^2+8=3^n.
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enescu
Thales


Joined: 20 May 2008
Posts: 171

PostPosted: Wed Jan 30, 2013 8:00 pm    Post subject: Reply with quote

Dacă n este par, n=2k, ecuaţia devine 8=(3^k-m)(3^k+m).
Obţinem imediat n=2, m=1.
Dacă n=2k+1, atunci, luānd m=2t+1 (evident, m trebuie sa fie impar), obţinem
4t^2+4t+8=3^{2k+1}-1=2\cdot (3^{2k}+3^{2k-1}+\ldots +3+1),
deci
2t^2+2t+4=3^{2k}+3^{2k-1}+\ldots +3+1,
imposibil, căci īn membrul drept avem suma unui număr impar de numere impare, care este un număr impar.
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