In part 1, I started a discussion on the ordered triples of integers whose squares form an arithmetic progression.

In other words,

, or equivalently

The solutions are of the form

for any integers and

The ordered triple is a primitive arithmetic progression triple if

If is a primitive arithmetic progression triple

so are

OR

OR

So, if is a primitive arithmetic progression triple

so are

Similarly, if is a primitive arithmetic progression triple

so are

Prove that if is a primitive arithmetic progression triple so are

are also a primitive arithmetic progression triples.

and recursively, for

… …

,

…………………………….

…………………………….

…………………………………………..

…………………………………………..

Using the An, Bn, Cn formulae doesnt generate squares in Ap, the results are in Ap with the same common difference as the original Ap though. I use the method to generate squares in Ap from Pythagorean triples. Lets look at the (3, 4, 5) PT, from this we can generate the (1, 25, 49) squares in Ap with difference 24, by this method.

1 = (4 – 3)^2

25 = 5^2

49 = (4 + 3)^2

or for the (12, 35, 37) PT generates

(529, 1369, 2209) or (23^2, 37^2, 47^2) CD 840.

This method works for all PT’s

Paul

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