=>12m^2 + 12m + 3 = 4n^2-1
=>3(2m + 1)^2 = (2n - 1)(2n + 1)
因(2n + 1),(2n - 1)是相差2的奇数,所以互素。因(2n - 1)(2n + 1)含有3和整数平方的因子,所以有
(1)2n - 1 = 3a^2, 2n + 1 = b^2 => b^2 = 3a^2 + 2.但整数平方/3只能余0,1.(1)不成立。
(2) 2n - 1 = a^2, 2n + 1 = 3b^2
因2n - 1是奇数,所以a是奇数。令a=2k+1,=> 2n - 1 =a^2=(2k+1)=4k^2 + 4k + 1=>2n =4k^2 + 4k + 2
=>n =2k^2 + 2k + 1=k^2 + (k^2+2k + 1)=k^2 + (k+1)^2