## Integer of the form 4xy(x + y)(x – y) divisible by 5

Find a number of the form   $4 \,xy \,(x + y) \,(x - y)$   which is divisible by 5,
the quotient being a square.

If we take x = 5, and y equal to a square such that   $(x+y)$   and   $(x-y)$   are also squares.
The least possible value for y is 4, in which case

$4 \,xy \,(x + y) \,(x - y) \; = \; 4\times 5\times 4\times 9\times 1 \; = \; 720 \; = \; 5 \times 12^2$

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Let’s look at ordered triples of integers   $(a, b, c)$   whose squares form an arithmetic progression.

In other words,
$b^2 \; - \; a^2 \; = \; c^2 \; - \; b^2$,     or equivalently
$2 \, b^2 \; = \; a^2 \; + \; c^2$

The solutions are of the form

$a \; = \; \pm k \, (m^2 \; - \; n^2 \; - \; 2 \, m \, n)$
$b \; = \; \pm k \, (m^2 \; + \; n^2)$
$c \; = \; \pm k \, (m^2 \; - \; n^2 \; + \; 2 \, m \, n)$

for any integers   $m$   and   $n$

The ordered triple   $(a, b, c)$   is a primitive arithmetic progression triple if   $k = 1$

$c^2 - b^2 = (m^2 - n^2 + 2 \, m \, n)^2 - (m^2 + n^2)^2 = 4 \, m n \, (m-n) \, (m+n)$
$b^2 - a^2 = (m^2 + n^2)^2 - (m^2 - n^2 - 2 \, m \, n)^2 = 4 \, m n \, (m-n) \, (m+n)$

$(m^2 - n^2 + 2 \, m \, n)^2 + (m^2 - n^2 - 2 \, m \, n)^2 = 2 \, (m^2+n^2)^2 = 2 \, b^2$

We would need to have:

$c^2 - b^2 \; = \; b^2 - a^2 \; = \; 5 \, v^2$

that is,    $4 \, m\, n \, (m-n) \, (m+n) \; = \; 5 \, v^2$

if   m = 5   and   n = 4,   then

a = -31,     b = 41,     c = 49

$b^2 - a^2 = 41^2 - 31^2 = 720$
$c^2 - b^2 = 49^2 - 41^2 = 720$

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See also Paul’s solutions at the bottom of the page

math grad - Interest: Number theory
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### 2 Responses to Integer of the form 4xy(x + y)(x – y) divisible by 5

1. paul says:

Here are a few, format is {x, y, 4 x y(x + y)(x – y), Sqrt[(4 x y(x + y)(x – y))/5]}

{5,4,720,12}
{9,1,2880,24}
{10,8,11520,48}
{15,12,58320,108}
{18,2,46080,96}
{20,16,184320,192}
{25,20,450000,300}
{27,3,233280,216}
{30,24,933120,432}
{35,28,1728720,588}
{36,4,737280,384}
{40,32,2949120,768}
{45,5,1800000,600}
{45,36,4723920,972}
{50,40,7200000,1200}
{54,6,3732480,864}
{55,44,10541520,1452}
{60,48,14929920,1728}
{63,7,6914880,1176}
{65,52,20563920,2028}
{70,56,27659520,2352}
{72,8,11796480,1536}
{75,60,36450000,2700}
{80,64,47185920,3072}
{81,9,18895680,1944}
{85,68,60135120,3468}
{90,10,28800000,2400}
{90,72,75582720,3888}
{95,76,93831120,4332}
{99,11,42166080,2904}
{100,80,115200000,4800}

Paul.

• benvitalis says:

Thanks. I posted a different solution