Conjecture About Squares of Consecutive Numbers

Date: 08/28/2001 at 22:06:58
From: Bryan
Subject: Squares of Consecutive Numbers

I noticed last year that when any two rational numbers whose absolute 
values are 1 apart are squared, the difference between the two 
resultant numbers is equal to the sum of the two original numbers. 

Example:

  4^2 - 3^2 = 4 + 3

  4.3^2 - 3.3^2 = 4.3 + 3.3

(with a negative number use absolute value)

  (-3)^2 - 2^2 = |-3| + |2|

  [(-3)^2] - [(-2)^2] = |-3| + |-2|
                            or
                        (-3) + (-2) since both numbers are negative

I was unable to find anything about it in the archives, possibly 
because I have no idea what to call it.

Date: 08/29/2001 at 09:06:05
From: Doctor Rick
Subject: Re: Squares of Consecutive Numbers

Hi, Bryan, thanks for writing to Ask Dr. Math.

I don't know that it has a name, but it can be proved in general by 
algebra. 

Let the first number be n; then the second number is (n+1). The 
difference of their squares is

 |(n+1)^2 - n^2|

 |n^2 + 2n + 1 - n^2|

 |2n + 1|

 |n + (n+1)|

which is the sum of the two numbers. Note that this is true not just 
for rational numbers, but for all numbers. 

I used the definition of "difference" as the greater number minus the 
lesser, which is the absolute value of (either number minus the 
other). Note that there is one difference between my result and your 
statement: you put the absolute values around each number separately:

  (n+1)^2 - n^2 = |n| + |n+1|

I put the absolute values around the sum:

  (n+1)^2 - n^2 = |n + (n+1)|

These are the same unless the numbers have different signs. Let's try 
n = -1/4 and n+1 = 3/4:

  |(-1/4)^2 - (3/4)^2| = |1/16 - 9/16|
                       = |-8/16|
                       = 8/16
                       = 1/2

  |n| + |n+1| = |-1/4| + |3/4|
              = 1/4 + 3/4
              = 1

so your statement isn't true in this case. On the other hand,

  |n + (n+1)| = |-1/4 + 3/4|
              = |1/2|
              = 1/2

The difference of the squares is equal to the absolute value of the 
sum, not the sum of the absolute values.

You were very observant to notice the pattern that the difference of 
the squares of consecutive numbers is the sum of the numbers. You had 
a good idea when you thought about extending the pattern to include 
negative numbers. You found a way that worked for the numbers you 
checked. Good work!

Your work was what we call "induction." You noticed a bunch of facts 
and made a conjecture: a pattern that seemed to fit the facts. You 
tried adding new facts - negative numbers - and found that you had to 
change your pattern slightly to account for all the new facts you 
tried. You were still using induction - an important part of math.

I have shown you the next step that a mathematician takes. The next 
step is "deduction": you try to PROVE that the pattern works for ALL 
numbers, using the rules of numbers. Just testing the pattern on lots 
of numbers doesn't prove that it works for ALL numbers. In fact, I 
have shown that your pattern doesn't work for n between -1 and 0. 

Using algebra to try to prove your conjecture, I ended up proving a 
modified theorem. Now we KNOW that this theorem is true for ALL 
numbers. It's no longer a conjecture.

Thanks again for writing. This has been an interesting exploration of 
the way math works, and I hope it is helpful to you. Keep up the good 
work!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/