Întrebare
Aratati ca 2011^2011 poate fi scris ca o suma de cinci patrate perfecte.
Întrebare a fost pusă de: USER4856
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72 Răspunsuri
Răspuns (72)
2011^2011=2011*2011^2010=(1936+64+9+1+1)*2011^2010=
= (44^2+8^2+3^2 +1+1)*2011^2010=
= 44^2 *(2011^1005)^2 +8^2* (2011^1005)^2+3^2*(2011^1005)^2+(2011^1005)^2+(2011^1005)^2=
=(44*2011^1005)^2 +(8*2011^1005)^2+(3*2011^1005)^2+(2011^1005)^2+(2011^1005)^2
= (44^2+8^2+3^2 +1+1)*2011^2010=
= 44^2 *(2011^1005)^2 +8^2* (2011^1005)^2+3^2*(2011^1005)^2+(2011^1005)^2+(2011^1005)^2=
=(44*2011^1005)^2 +(8*2011^1005)^2+(3*2011^1005)^2+(2011^1005)^2+(2011^1005)^2