Subject: math theory on consecutive squares

Dear Q and Q:

As a teacher at a school called Educere in Houston, I have a ninth-grade student who discovered the following shortcut last year as an eight-grader. I promised him I would find out if it had ever been published under a mathematician's name, and if not, how to go about getting it published under his name, such as the "Ben Rose" rule of consecutive squares.

What he noticed is that given any two consecutive integers (or n and n+1 for any rational number greater than or equal to 2), the difference between their squares was equal to the sum of the two numbers.

i.e., comparing 2 and 3: the difference between 4 and 9 is 5, or 2+3 comparing 4 and 5: the difference between 16 and 25 is 9, or 4+5 etc.

in general, the difference between consecutive squares:

   (n+1)2 - n2
   = n2 +2n +1 - n2
   = 2n+1
   = n + (n+1),
the sum of the two numbers.

Again, our question is:

  1. has this shortcut or mathematical trick ever been published?
  2. if so, by which mathematician under which name, and where can we find it?
  3. if not, how does one go about publishing and naming such a mathematical rule?
Thank you for your assistance and attention to this student-teacher query.

Yours truly,

Emily Nghiem
Ben Rose

Hi Emily and Ben,

This is a nice observation. It was probably known to the ancient Greeks, perhaps before Pythagoras (500 BC). Their observation would have been based on the fact that the sum of consecutive odd numbers is a perfect square:

1, 1+3=4, 1+3+5 = 9, 1+3+5+7=16 etc.

All the squares can be obtained this way and thus the difference between consecutive squares is always an odd number, and (as Ben noticed) any odd number is the sum of consecutive integers.
   Another algebraic proof similar to yours is to use that fact that a2 = b2 = (a-b)(a+b). Thus
   (n+1)2 - n2
   = (n + 1 - n)(n + 1 + n)
   = 1(2n + 1)
   = n + (n+1),

The fact that Ben was not the first person to make this observation doesn't take anything away from his achievement. He's doing what students and mathematicians are supposed to do!

Chris and Penny

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