S:E18.128: Difference between revisions

Revision as of 12:18, 30 September 2023

S:E18.128 (Vasile Ienuțaș)

Scrieți numărul $2018^{2017}$ ca sumă de patru pătrate perfecte nenule distincte.

Soluție.

Din $2018=3^{2}+4^{2}+12^{2}+43^{2}$ se obține

2018^{2017}=2018\cdot 2018^{2016}=\left(3^{2}+4^{2}+12^{2}+43^{2}\right)\cdot \left(2018^{1008}\right)^{2}

Deci

2018^{2017}=\left(3\cdot 2018^{1008}\right)^{2}+\left(4\cdot 2018^{1008}\right)^{2}+\left(12\cdot 2018^{1008}\right)^{2}+\left(43\cdot 2018^{1008}\right)^{2}

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 '''Soluție.'''
+Din <math>2018 = 3^2+4^2+12^2+43^2</math> se obține
+<math display="block">2018^{2017} = 2018 \cdot 2018^{2016} = \left ( 3^2 + 4^2 + 12^2 + 43^2 \right ) \cdot \left( 2018^{1008}\right)^2  </math>Deci
+<math display="block">2018^{2017} = \left( 3 \cdot 2018^{1008}\right)^2 + \left( 4 \cdot 2018^{1008}\right)^2 + \left( 12 \cdot 2018^{1008}\right)^2 + \left( 43 \cdot 2018^{1008}\right)^2</math>