Question

若m,n是整数,且n2+3m2n2=30m2+517,求3m2n2.

Answer 1

n^2+3m^2n^2=30m^2+517,

n^2(1+3m^2)=30m^2+517

等式右边为奇数,要等式成立,则等式左边n^2,3m^2+1应均为奇数,n^2为奇数,m^2为偶数.

n^2=(30m^2+517)/(3m^2+1)=(30m^2+10+507)/(3m^2+1)=10+507/(3m^2+1)

507≥3m^2+1

m^2≤12

又m为偶数,只有m^2=4时,n^2=49,均为整数.

3m^2n^2=3×4×49=588