E:16910

E:16910 (Teodora Zetea & Bogdan Zetea)

Aflați soluțiile întregi ale ecuației $x^{4}+4y^{4}=3796.$

Soluție

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 x^4 + 4y^4 = \left(x^2+2y^2\right)^2 - 4x^2y^2 = \left(x^2+2y^2-2xy\right)\left(x^2 + 2y^2 +2xy\right),} ecuația dată revine la 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 \left(x^2+2y^2-2xy\right)\left(x^2 + 2y^2 +2xy\right) = 3796.} Din 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 2y^2-2xy\, \vdots \, 2} 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 2y^2+2xy\, \vdots \, 2} se deduce că expresiile pozitive 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 x^2 + 2y^2-2xy} 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 x^2 +2y^2 +2xy} au aceeași paritate.

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 3796 = 2^2 \cdot 13\cdot 73} , sunt posibile situațiile 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 \begin{cases} x^2+2y^2-2xy = 26 \\ x^2 + 2y^2 +2xy = 146 \end{cases}} 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 \begin{cases} x^2+2y^2-2xy = 146 \\ x^2 + 2y^2 +2xy = 26. \end{cases}} Se obțin soluțiile <math>\left(x,y\right)\in \left\{ \left(\pm 6,-5\right), \, \left(\pm 6, 5\right) \right\}<.math>