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 UN cargo (Posted on 2013-03-13)
A triplet (1,25,49) represents 3 perfect squares forming an arithmetic progression.

1. Provide few additional samples like the above.
2. What can be said about the possible values of d (the constant
difference between the adjacent members of an arithmetic progression)?
3. How relates the title of my post to its contents?

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 Possible solution | Comment 1 of 9

This answer deals only with 'regular' solutions.

Let the 3 numbers to be squared be a, b, c. By basic algebra:

a^2+d=b^2
a^2+2d=c^2
From 1
a^2+2(b^2-a^2)=c^2
2b^2=c^2+a^2

This will always be true when a=b=c, but assume a and b are different.

a can be any integer we choose, and there will always be a solution:
5a=b, 7a=c, d=(5^2-1)a^2
29a=b, 41a=c, d=(29^2-1)a^2
169a=b, 239a=c, d=(169^2-1)a^2, etc.

Generally, let x be a number such that 2x^2-1=y^2 has a solution in the positive integers. Then a, b=(ax), c=(ay), d=(x^2-1)a^2 is a solution:

(ax)^2-a^2 = (ay)^2-(ax)^2
a^2(x^2-1) = a^2(y^2-x^2)

Clearing the a^2 terms and collecting the others: 2x^2-1=y^2;  the relationship between x and y is independent of a and is fixed.

As I said above, 'regular' solutions, because a might, e.g. be 7, in which case {b,c,d} = {13,17,120}{17,23,240} etc. also represent valid, but not 'regular' solutions.

I'm not sure about the 'UN cargo' clue, though. If it is a clue, that is...

Edited on March 13, 2013, 12:21 pm
 Posted by broll on 2013-03-13 12:11:52

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