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 Classical Rules 2 (Posted on 2006-04-20)
Find three squares such that each minus the product of the three gives a square.

Classical Rules: Let a "square" be any number that is the square of a rational number.

 No Solution Yet Submitted by goFish Rating: 4.0000 (1 votes)

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 re: Partial Solution | Comment 2 of 3 |
(In reply to Partial Solution by tomarken)

I first tried to see if there were any solutions where the three squares were equal other than (1,1,1)

If each is rational (say a/b) then

(a/b)^2 - (a/b)^6 = (c/d)^2 where c/d is rational.

This simplifies to (b^4-a^4) = [((b^3)*c)/(ad)]^2 in which case we know from Fermat's Last Theorum that the difference of powers to the fourth cannot equal a square among integers, and since the difference is a non-zero integer (remember I am excluding (1,1,1) its square root cannot be rational without it being a square, which it isn't. Therefore at least there is no other solution with three equal squares.

Edited on April 20, 2006, 10:51 pm
 Posted by Eric on 2006-04-20 22:19:40

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