Prove that there is a positive integer that can be written as the sum of squares of positive integers in two different ways. For some reason this is not clicking in my head and I can not figure it out. Any help will be greatly appreciated and I would appreciate no straight forward answer since I am trying to understand.

There exist many such examples.But the best example is the Charamichael number 1105.It can be expressed as the sum of squares of two integers in 4 different ways or so.

In general let the statement that the numbers be x,y and z which satisfy:
x^2 + y^2 = z
be true.

If x and y are co-prime,i.e their HCF is 1,one of them is divisible by 3,then the above property satisfies for certain numbers like

13^ + 6^2 = 205 = 3^2 + 14^2
15^2 + 8^2 = 289 = 17^2 + 0^2 (But here zero IS NOT a positive integer)