Bizarre Square Numbers

Back to the time when I was in class VI, there was a challenge between a classmate and myself to memorise the squares of first fifty natural numbers. And I worked like mad on it. I memorised it all, and thanks to the effort, I still remember them.

                But I found out something. This, as I see, has always been overlooked. I observed a pattern to generate the next square number from the previous ones. I thought it was very cool to have it. Since I had prepared the list only for first fifty square numbers, I manually verified it till that point.

                9 years later.
                I was about to complete my second year in college. I had that feeling that the pattern must be true for more than 50 numbers; I could not convince myself why it should be that the pattern was true for the first 50 numbers and not for the rest of them. So, I checked manually for some more. Till about 87 probably.
                It wasn’t a point trying to check the numbers manually. And checking it till a fixed number does not guarantee that the pattern is correct. In order to facilitate my work, I decided to find out a function for it, and check it via. computer programs. So, I set out for it, and after some work done, I thought I had a good enough flow chart.

                Things were set right. I requested my brother to prepare a program for it, since I don’t have any worthy knowledge of programming as such. But the same question kept ringing over. The program can also check for a range depending upon the processor.

                The only thing left was to prove the result in general for any number. I set out for it. It took me nearly two days to find out that there was a way in which I could write a recursive relation as an explicit relation. Thanks to the Binet’s formula for Fibonacci sequence that gave me a clue. I prepared the general function, all explicit.

                And then verified the result for each of the ten (depending on units digits) types of square numbers. And then I came to my favourite result- HENCE PROVED.

                I surfed the net for some time to see if any similar works existed. I got none (till now). I think I should publish it.

Note:- The result is not a very familiar type of recursive relation (examples of which you can find on the Wikipedia page for Square numbers). It employs a ‘modulus’ function, and is an attempt to study square numbers as a combination of its unit digit and the rest of the number.