Find all integers m that can make m ^ 2 + m + 7 a complete square number RT

Find all integers m that can make m ^ 2 + m + 7 a complete square number RT

Let m ^ 2 + m + 7 = k ^ 2, so (M + 1 / 2) ^ 2 + 27 / 4 = k ^ 2, so (M + 1 / 2) ^ 2-k ^ 2 = - 27 / 4, so (M + 1 / 2 + k) (M + 1 / 2-k) = - 27 / 4, so [(2m + 2K + 1) / 2] [(2m-2k + 1) / 2] = - 27 / 4, so (2m + 2K + 1) (2m-2k + 1) / 4 = - 27 / 4, so (2m + 2n + 1) (2m-2k + 1) = - 27, because k > 0

If 20m is a positive integer, then the minimum value of the positive integer m is______ ．

∵ 20m is a positive integer, according to the meaning of the question, 5m is the smallest complete square, so the answer is 5