S:E18.131

S:E18.131 (Nicolae Mușuroia)

Determinați cel mai mic număr natural pătrat perfect care se poate scrie ca sumă de 2018 numere naturale consecutive.

Soluție

Fie 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 k} numărul căutat. Atunci

$$x+{\bigl (}x+1{\bigr )}+\ldots +{\bigl (}x+2017{\bigr )}=k^{2}$$

ceea ce revine, în mod echivalent, la

$$1009\cdot {\bigl (}2x+2017{\bigr )}=k^{2}$$

Deci $1009|k^{2}$, iar cum 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 1009} este număr prim, se deduce că 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 1009 | k} .

Atunci, există 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 l \in \mathbb{N}^\ast} , cel mai mic posibil, pentru care 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 k=1009 \cdot l} .

Se obține 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 2x=1009\cdot l^2 - 2017 \in \mathbb{N} } , de unde rezultă 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 l=3 } și

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 k=3027 }