14440

14440 (Vasile Ienuțaș și Radu Pop)

Se consideră numărul natural $A=a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+.....+a_{2012}^{2}$ unde $a_{1},a_{2},a_{3},.....,a_{2012}$ sunt numere prime, mai mari sau egale cu 5. Arătați că $B=2\cdot A+2013$ nu este pătrat perfect.

Soluție:

Dacă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i} , cu Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in \{1,2,3,\ldots,2012\}} , este număr prim mai mare decât 5, atunci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i^2 \equiv M8 + 1} , pentru orice Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in \{1,2,3,\ldots,2012\}} . Prin urmare, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = 2 \cdot \left(\mathcal{M}8 + 2012\right) + 2013 = \mathcal{M}8 + 6037 = \mathcal{M}8 + 5} , care nu este pătrat perfect.