證：
n=1時,x²-y²=(x+y)(x-y),包含因式x+y,能被x+y整除.
假設(shè)當n=k(k∈N+且k≥1)時,x^(2k)-y^(2k)能被x+y整除,則當n=2(k+1)時,
x^[2(k+1)]-y^[2(k+1)]
=[x^(2k+1)-y^(2k+1)](x+y)-yx^(2k+1)+xy^(2k+1)
=[x^(2k+1)-y^(2k+1)](x+y)-xy[x^(2k)-y^(2k)]
[x^(2k+1)-y^(2k+1)](x+y)中包含因式x+y,能被x+y整除；xy[x^(2k)-y^(2k)]中包含能被x+y整除的因式x^(2k)-y^(2k),能被x+y整除.即當n=k+1時,x^[2(k+1)]-y^[2(k+1)]能被x+y整除.
綜上,得x^(2n)-y^(2n)能被x+y整除.