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IMAC Juniori II 15 mai 2010 Subiectul IV

 
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Andi Brojbeanu
Newton


Joined: 22 Mar 2009
Posts: 383
Location: Targoviste (Dambovita)

PostPosted: Tue Jun 15, 2010 8:31 pm    Post subject: IMAC Juniori II 15 mai 2010 Subiectul IV Reply with quote

Sa se determine toate numerele naturale p si q stiind ca 1+2\cdot p^q este patrat perfect, iar p este numar prim.
Romania
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Brojbeanu Andi Gabriel, clasa XII-a
Colegiul National "Constantin Carabella" Targoviste
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seby97
Euclid


Joined: 04 Aug 2011
Posts: 31

PostPosted: Thu Aug 25, 2011 9:14 am    Post subject: Reply with quote

Fie a^2=1+2p^q de unde (a-1)(a+1)=2 p^q de unde a trebuie sa fie impar,a>1
Daca a=>3 avem ca (a-1)(a+1) este multiplu de 4 de unde deducem ca p^q este multiplu de 2 si cum p=prim rezulta p=2
Ecuatia data devine: 1+2^(q+1)=a^2.Notam a=2b+1 de unde
2^(q+1)=2b(2b+2) adica 2^(q-1)=b(b+1).b=1 este solutie de unde q=2
Pentru b>2 avem ca b(b+1) mai are si un numar impar in descompunere si nu poate fi 2^x
Deci a=3,p=2,q=2
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