Doing Math In My Head, (instead of sleeping) Again.

It occurred to me that for every square, say 30 x 30, you can get to the next one, 31 x 31 by simply adding the first and the second numbers on.  So 31 squared is 30 squared + 30 +31.  It becomes more obvious when you think of a grid:  30 rows x 30 columns will go to 31 rows and 31 columns by adding a row of 30 and then a column of 31.

It thus follows that if any two consecutive numbers add to a perfect square, then the three could form a right triangle of whole numbers.  Example:  12 + 13 = 25 which is 5 squared, so 12, 13 and 5 will make up the sides.  It must be the case, since you get from 12 squared (144) to 13 squared (169) by adding 5 squared (25).

Just in my head, I found one which would have been too high to calculate:  112 + 113 = 225 and the square root of 225 is 15.

I wonder if there's a name for this pattern of consecutive numbers and then a third number, which together make a perfect square.

I asked GPT about this and while there doesn't seem to be a name for this, it was known in ancient times.  It works for any odd square because there will always be two consecutive numbers which add to this value.  It obviously cannot work for even squares, since you can't add consecutive integers and reach an even number.

You can go as high as you want! 6,160, 111, 6,161 works, but you may want to check with a calculator!