2007 UNCO Math Contest II Problems/Problem 6

Problem

(a) Demonstrate that every odd number $2n+1$ can be expressed as a difference of two squares.

(b) Demonstrate which even numbers can be expressed as a difference of two squares.

Solution

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(a) For any odd number $2n+1$ :

$2n+1 = n^2 + 2n +1 - n^2$

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

(b) When expressing an even number as a difference of two numbers, these must be of the same parity, i.e. both must be either even or odd. The parity of a number stays the same when squaring it. This gives us two cases:

Case 1: 2n is the difference of two even numbers:

$2n = (2m)^2 - (2p)^2$

$2n = 4(m^2 - p^2)$

Therefore 2n is a multiple of 4.

Case 2: 2n is the difference of two odd numbers:

$2n = (2m+1)^2 - (2p+1)^2$

$2n = 4m^2 + 4m + 1 - 4p^2 - 4p - 1$

$2n = 4(m(m+1) - p(p+1))$

Therefore 2n is a multiple of 4.

For both cases we find that only even numbers which are multiples of 4 can be expressed as a difference of two squares.

2007 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2007 UNCO Math Contest II Problems/Problem 6

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