I tried coming up with a proof, but instead I have a new observation I can't explain: FACT: No prime number can end in an even number (because all such numbers

I tried coming up with a proof, but instead I have a new observation I can't explain: FACT: No prime number can end in an even number (because all such numbers are divisible by 2) or in 5 (because all such numbers are divisible by 5). So, all prime numbers must end in: 1, 3, 7, or 9. FACT: When you multiply 2 numbers, the units digit of the product is the same as the units digit of the product of the units digits of the two numbers. FACT: Combining the last two facts, every prime number squared must end in 1 (if the prime ended in 1 or 9) or 9 (if the prime ended in 3 or 7). FACT: Thus, every prime square modulo 40 must be one of the following numbers (and cannot be any other number): 1, 9, 11, 19, 21, 29, 31, or 39. So, I just have to figure out now why 6 of those 8 numbers can't be prime squares modulo 40. That's when I noticed something odd: no number x, such that x modulo 40 is 11, 19, 21, 29, 31, or 39, is a perfect square. (Tried this through x = 1 million.) Anyone know why that is?