Question

cum demonstram ca 4k+2 nu este patrat perfect

Answer 1

Presupunem ca exista n²=4k+2

fie n∈{4m, 4m+1, 4m+2, 4m+3}, in functie de restul impartirii la 4

1. n=4m

=> n²=(4m)²=16m²=M4=4k

2. n=4m+1

=> n²=(4m+1)²=16m²+8m+1=M4+1=4k+1

3.n=4m+2

n²=(4m+2)²=16m²+16m+4=M4=4k

4. n=4m+3

=>n²=(4m+3)²=16m²+24m+9=4(4m²+6m+2)+1=M4+1=4k+1

=> n² poate avea forma 4k sau 4k+1

=> 4k+2 nu poate fi patrat perfect