DIFFERENCE OF SUCCESSIVE SQUARES

I was on a long drive recently and got bored after a couple of hours. At times like these, I start to read license plates to check for interesting combinations of letters, prime numbers, etc.

I noticed that the number 361 was popping up more frequently than one would expect by mere chance. Other than being the cubic inch displacement of a Chrysler V-8 from some years back, the number seemed to have no special significance. I did some mental division to see what its prime factors were and found that it is 19 squared. Mentally dividing by 19 is not a simple thing to do for most of us. The next square is 400, the square of 20. This one is very easy to calculate mentally. I wondered if there were some trick to work from the easy square of 20 to get the difficult square of 19.

I took the difference between the squares, 400 - 361 = 39. After a bit of thought, I suddenly realized that 39 = 19 + 20, the sum of the two numbers I was squaring. The proof that this is the general rule was easy to work out and follows below:

1. Take any two successive positive integers. They can be represented as X and X+1.
2. Square each, yielding X^2 and (X+1)^2 = X^2 + 2X + 1.
3. Take the difference of the squares: (X^2 + 2X + 1) - X^2 = 2X + 1
4. Expand the difference algebraically and group terms: 2X + 1 = X + X + 1 = X + (X+1)
5. But X + (X+1) is the sum of the two successive positive integers we started with. Hence, The difference between the squares of two successive positive integers is the sum of those two integers. QED

Had I known this trick, I could have computed the square of 19 by taking the square of 20 and subtracting 39. Trying a few more,
    Start with 5,6; squares are 25 and 36; difference is 11, which is 5+6
    Start with 10,11; squares are 100 and 121; difference is 21, which is 10+11
    Start with 123,124; squares are 15,129 and 15,376; difference is 247, which is 123+124

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