E.570. a) Arătați că 10210^2102 se poate scrie ca sumă a patru cuburi perfecte. b) Determinați numerele naturale nenule distincte m,n,p,q,m,n,p,q,m,n,p,q, știind că m3+n3+p3+q3=102021.m^3+n^3+p^3+q^3=10^{2021}.m3+n3+p3+q3=102021.
Răspuns: a) 102=43+33+23+13;10^2=4^3+3^3+2^3+1^3;102=43+33+23+13; b) Putem alege: m=4⋅10673, n=3⋅10673, p=2⋅10673, q=10673.m=4 \cdot 10^{673},~ n=3 \cdot 10^{673},~ p=2 \cdot 10^{673},~ q=10^{673}.m=4⋅10673, n=3⋅10673, p=2⋅10673, q=10673.
a) 102=64+27+8+1=43+33+23+13.10^2=64+27+8+1 = 4^3+3^3+2^3+1^3.102=64+27+8+1=43+33+23+13.
b) 102021=102⋅102019=10^{2021} = 10^2 \cdot 10^{2019}=102021=102⋅102019= =(43+33+23+13)⋅102019==(4^3+3^3+2^3+1^3) \cdot 10^{2019}==(43+33+23+13)⋅102019= =43⋅(10673)3+33⋅(10673)3+23⋅(10673)3+(10673)3.=4^3 \cdot (10^{673})^3 + 3^3 \cdot (10^{673})^3 + 2^3 \cdot (10^{673})^3 + (10^{673})^3.=43⋅(10673)3+33⋅(10673)3+23⋅(10673)3+(10673)3.
Deci putem alege: m=4⋅10673, n=3⋅10673, p=2⋅10673, q=10673.m=4 \cdot 10^{673},~ n=3 \cdot 10^{673},~ p=2 \cdot 10^{673},~ q=10^{673}.m=4⋅10673, n=3⋅10673, p=2⋅10673, q=10673.