67,  146, 363, 10003 are not perfect squares as they leave remainder 3,2,3 and 3 respectively, when divided by 4,

Explanation

The perfect square reminder of `(0,1)`
`(4x+1,4x)`
where `(3)` is natural number
`(67,146,363,10003)`are not perfect square
Because the reminder of `146` is `2`and the reminder of `67,363,10003` is `3` which is not in the form
`(4x+1,4x)`