2n+1 patrat perfect
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Radu Titiu
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2n+1 patrat perfect
Daca \( 2n+1 \) este patrat perfect atunci aratati ca \( n+1 \) se poate scrie ca o suma de doua patrate perfecte , iar \( 3n+1 \) se poate scrie ca o suma de trei patrate perfecte.
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Laurian Filip
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\( 2n+1=a^2 \)
evident a - impar .
Fie k astfel incat \( a=2k+1 \)
\( 2n+1=(k+k+1)^2 \)
de unde \( 2n=2k(2k+2) \) adica \( n=2k(k+1) \)
\( n+1=2n+1-n=(k+k+1)^2-2k(k+1)=k^2+2k(k+1)+(k+1)^2-2k(k+1)=k^2+(k+1)^2 \)
\( 3n+1 \) chiar cred ca nu se poate scrie ca suma de 3 patrate perfecte... in cazul in care voiai sa scri \( 3n+2 \) atunci \( 3n+2=a^2+k^2+(k+1)^2 \)
evident a - impar .
Fie k astfel incat \( a=2k+1 \)
\( 2n+1=(k+k+1)^2 \)
de unde \( 2n=2k(2k+2) \) adica \( n=2k(k+1) \)
\( n+1=2n+1-n=(k+k+1)^2-2k(k+1)=k^2+2k(k+1)+(k+1)^2-2k(k+1)=k^2+(k+1)^2 \)
\( 3n+1 \) chiar cred ca nu se poate scrie ca suma de 3 patrate perfecte... in cazul in care voiai sa scri \( 3n+2 \) atunci \( 3n+2=a^2+k^2+(k+1)^2 \)