  We know that any prime number of form $4s+1$ can be written as a sum of two
perfect square numbers. As a consequence of Goldbach's weak conjecture, any
number great than $10$ can be represented as a sum of four primes. We are
motivated to consider an equation with some constraints about digital sum for
the four primes. And we conclude that the square root of the digital sum of the
four primes will greater than $4$ and will not be a multiple of $3$ if the
equation has solutions. In the proof, we give the method of determining whether
a number is a perfect square.
