Question

知x-y=1,x^3-y^3=4,x^13-y^13=?

Answer 1

x^3-x^3=(x-y)(x^2+xy+y^2)=4
那么(x-y)^2+3xy=4
整理即:xy=1
x^13-y^13=(x-y)(x^12+x^11*y^2+x^10*y^3....+x^6y^6.......+y^12)
把x-y带入
那么有x^13-y^13=x^12+x^11*y^2+x^10*y^3....+x^6y^6.......+y^12

再把xy带入,那么有x^13-y^13=x^12+x^10+x^8+x^6+...1+y^2+y^4.....y^12
而x^2+y^2=(x-y)^2+2xy=3
x^4+y^4=(x^2+y^2)^2-2x^2y^2=3^2-2=7
x^6+y^6=(x^2+y^2)(x^4+y^4)-x^2y^2(x^2+y^2)=3*7-3=18
x^8+y^8=(x^4+y^4)^2-2x^4y^4=7^2-2=47
x^10+y^10=(x^4+y^4)(x^6+y^6)-x^4y^4(x^2+y^2)=123
x^12+y^12=(x^6+y^6)^2-2x^6y^6=322
加起来就是x^13-y^13的解
最后算得:521