$m^2 - n^2 = N$

Number Theory Level 2

Which of the following can be written as a difference of two perfect squares?

605 All of these 88 847 180

1 solution

Munem Shahriar
Sep 27, 2017

$44^2 - 33^2 = 847$
$33^2 -22^2 = 605$
$13^2 - 9^2 = 88$
$18^2 - 12^2 = 180$

Hence the answer is $\color{#20A900} \boxed{\text{all of these}}$

Is there any method or formula for questions like this? @Brian Charlesworth @Calvin Lin @Geoff Pilling

Kaushik Chandra - 3 years, 7 months ago

Noting that $m^{2} - n^{2} = (m - n)(m + n)$ , an integer $N$ can be written as the difference of two squares if it can be factored as a product of two even numbers or two odd numbers.

Suppose $N = ab$ , where both $a,b$ are even and $a \lt b$ . Then let $a = m - n$ and $b = m + n$ . Then $a + b = 2m \Longrightarrow m = \dfrac{a + b}{2}$ , (which is an integer as $a,b$ are even), and $n = \dfrac{b - a}{2}$ , (again guaranteed to be an integer). So for example, $88 = ab$ with $a = 4, b = 22$ , giving us $m = 13, n = 9$ . (We could also have $a = 2, b = 44$ giving us $m = 23, n = 21$ .)

Similarly where both $a,b$ are odd. For example, $847 = 7 \times 121$ , giving us $m = 64, n = 57$ .

Brian Charlesworth - 3 years, 7 months ago

Integers that cannot be written as the difference of two squares are $1, 2, 4, 6, 10, 14, 18, ....$ . Any odd integer $2k + 1, k \ge 1$ can be written as $(k + 1)^{2} - k^{2}$ . Any even integer of the form $4k, k \ge 2$ can be written as $(k + 1)^{2} - (k - 1)^{2}$ . No integer that is equivalent to $2$ mod $4$ can be written as the difference of two squares because the difference of two perfect squares is either $0,1$ or $3$ mod $4$ , (since perfect squares are all either $0$ or $1$ mod $4$ ).

Brian Charlesworth - 3 years, 7 months ago

@Brian Charlesworth – Note that 0 is a perfect square, so $1 = 1 - 0 , 4 = 4 - 0$ .

The characterization is that an integer $N$ can be written as the difference of 2 squares if and only if $N \not \equiv 2 \pmod{4}$ . This is explained by Brian above
1. why $4k+2$ doesn't work,
2. provided a construction for all the other cases.

Calvin Lin Staff - 3 years, 7 months ago

Since all odd numbers can be written as the difference of two perfect squares, I knew that the only possible answer was "All of these"

Geoff Pilling - 3 years, 7 months ago

m 2 − n 2 = N m^2 - n^2 = N m 2 − n 2 = N

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$m^2 - n^2 = N$