588.
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