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
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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.

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