Integers (a,b,c) whose squares form an arithmetic progression — Part 1

 
 
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 \; = \; D^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

D \; = \; L \, s^2,   for   L < 10

and where   (m, n) = 1

 

although, we still are getting multiples of primitive solutions, as shown in the table below:

 
SQR in AP

 

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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