证：
n=1时,x²-y²=(x+y)(x-y),包含因式x+y,能被x+y整除.
假设当n=k(k∈N+且k≥1)时,x^(2k)-y^(2k)能被x+y整除,则当n=2(k+1)时,
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.即当n=k+1时,x^[2(k+1)]-y^[2(k+1)]能被x+y整除.
综上,得x^(2n)-y^(2n)能被x+y整除.