1. (a) Show that the equation 4x=y^2 +z^2 +1 has no integer solutions. (b) Find all

Let integers a + b = x. Let y be odd then z is even.

Then 4a = y^2 +1 and 4b = z^2
Since z^2 has a factor of 4 then to have an integer solution 4a = y^2 + 1

And, any odd integer squared + 1 is not dividable by 4..

For integers 1, 3, 5, 7, 9,, 11; y^2+1 = 2, 10, 26, 50, 82, 121,

For Proof: Let y be odd and assume y^2 +1 not divisible by 4.

Therefore (y+2) ^2 + 1= (y^2+1) + 4y + 4.

The terms 4y + 4 are divisible by 4 and the sum of any number divisible by 4 added to a number not divisible by 4 is not divisible by 4.

There are an infinite number of non integer solutions.