Question

If (x,y,z) is a primitive Pythagorean triple, prove that z= 4k+1

Answer 1

If   is a primitive Pythagorean triple the and  

Since every integer is either even or odd so three case may arise both are even or both are odd or one of is even and another is odd .

Case 1 : If both x,y is even .

Then , for some

a contradiction to .

Case 2 : If both x , y are odd .

for some

is of the form

2 divides but 4 does not divided .

As 2 divides

a contradiction to 4 does not divides .

So both cannot be even and also both cannot be odd .

one of is odd and another is even .

Without loss of generality assume that is odd and is even .

   for some

,[ where s=k²+m²+1 ]

Hence is of the form .

Hence