More thoughts on numbers.

While driving to work the other day I had more abstract mathematical theorems pop into my head.

This time it was if there was a quick test to see if any given multiple digit number could have a integer for a square root.

Well, it seems there is a quick test. If the last digit of the number can be had by squaring any single digit number, then yes, it could have a integer as a square root.

Some examples:

<----n9 can have a integer square root, because 3^2 and 7^2 both have squares that end in 9 ( 9 and 49 ).
<----n3 cannot have an integer square root, because there is no n^2 that ends in 3.

The digits that can be acquired by squaring single digit numbers are 0, 1, 4, 5, 6, and 9.

Note; though that any number ending in 0 have to be <--n00 numbers and then the number has to be a square of a single digit number; i.e. 100, 400, 900, 1600 are fine, but 200, 300, 1500, 2000 do not work.

Any number that ends in 5 might have an integer square root because <-n5^2 will always end in <---n5.

An interesting side note is that the difference between any two adjacent integer squares is an odd number:
0 -> 1 (1); 1 -> 4 (3); 4 -> 9 (5); 9 -> 16 (7), etc.

Again this lion has too much time on his paws when driving to/from work. :=/