I think you misunderstand what is the halting problem. It's being able to tell whether a program will halt or not, for all conceivable programs. A human brain certainly can't do that.

For example, does this program halt? (Let's assume infinite-precision numbers, for simplicity. After all, a Turing machine can access an infinitely long tape.)

    for (int n = 3; ; n++)
      for (int a = 1; a < n; a++)
        for (int b = 1; b < n; b++)
          for (int c = 1; c < n; c++)
            for (int m = 3; m < n; m++)
              if (pow(a, m) + pow(b, m) == pow(c, m)) exit(1);

Show me that this program never halts, and you just proved Fermat's last theorem.

Edit: added one missing loop