求证：m,n都是大于1的整数时,m^4+4n^4一定是合数.

题目

求证：m,n都是大于1的整数时,m^4+4n^4一定是合数.

答案

m^4+4n^4 = m^4 + 4m^2n^2+4n^4 - 4m^2n^2 = (m^2+2n^2)^2 - 4m^2n^2
= (m^2-2mn+2n^2)(m^2+2mn+2n^2)
所以 m^4+4n^4一定是合数.