no integral solution

1) from a^2 + 2b = 7, let a = 2A + 1, we have 2A2 + 2A+ b = 3, futher let b = 2B +1, we then have A2 + A + B = 1;

2) from b^2 + 4c= -7 we can have B^2 + B + c = -2, then let c = 2C, we have B^2+B+2C = -2;

3) from c^2 + 6a = 14, we have C^2 + 3A = 2. consider modula 3, there is no integral solution for C.