E.568. a) Calculați 132+252+352.13^2+25^2+35^2.132+252+352. b) Arătați că numărul 201920192019^{2019}20192019 se poate scrie ca sumă a trei pătrate perfecte.
Răspuns: a) 2019;2019;2019; b) 20192019=(13⋅20191009)2+(25⋅20191009)2+(35⋅20191009)2.2019^{2019}=(13 \cdot 2019^{1009})^2 + (25 \cdot 2019^{1009})^2+ (35 \cdot 2019^{1009})^2.20192019=(13⋅20191009)2+(25⋅20191009)2+(35⋅20191009)2.
a) 132+252+352=13^2+25^2+35^2 =132+252+352= =169+625+1225=2019.=169+625+1225 = 2019.=169+625+1225=2019.
b) 20192019=2019⋅20192018=2019^{2019}=2019 \cdot 2019^{2018}=20192019=2019⋅20192018= (132+252+352)⋅20192018=(13^2+25^2+35^2) \cdot 2019^{2018}=(132+252+352)⋅20192018= =132⋅(20191009)2+252⋅(20191009)2+352⋅(20191009)2==13^2 \cdot (2019^{1009})^2 + 25^2 \cdot (2019^{1009})^2 + 35^2 \cdot (2019^{1009})^2==132⋅(20191009)2+252⋅(20191009)2+352⋅(20191009)2= (13⋅20191009)2+(25⋅20191009)2+(35⋅20191009)2.(13 \cdot 2019^{1009})^2 + (25 \cdot 2019^{1009})^2+ (35 \cdot 2019^{1009})^2.(13⋅20191009)2+(25⋅20191009)2+(35⋅20191009)2.