Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
High School Number Theory

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search