Question

1. a) Scrieti numarul 29 la puterea 2n+1 ca suma de doua patrate perfecte nenule.
b) Scrieti numarul 38 la puterea 2n+1 ca suma de trei patrate perfecte nemule.

2. Sa se arate ca N=10*2^(6n+2)+3*2^(6n+3) , unde n numat natural , este atat patrat perfect cat si cub perfect.

Answer 1

1.a)
29^(2n+1)=29*29^2n=(25+4)*29^2n=(5^2+2^2)*(29^n)^2=
=(5*29^n)^2+(2*29^n)^2
b)
38^(2n+1)=38*38^2n=(4+9+25)*38^2n=(2^2+3^2+5^2)*(38^n)^2=
=(2*38^n)^2+(3*38^n)^2+(5*38^n)^2

2.
N=10*2^(6n+2)+3*2^(6n+3)
N=2^(6n+2)*(10+3*2)
N=[2^(3n+1)]^2*16
N=[2^(3n+1)*4]^2, deci N e p.p.
N=[2^3n*2*4]^2=[2^(3n+3)]^2
N={[2^(n+1)]^3}^2
N={[2^(n+1)]^2}^3, deci N e si cub perfect