How many solution does the equation have in the set of ?

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- Sep 24th 2010, 11:38 PMjames_bondNumber of solutions of x^2 + xy + y^2 = 27 in Q
How many solution does the equation have in the set of ?

- Sep 25th 2010, 04:46 AMOpalg
- Sep 26th 2010, 11:48 AMjames_bond
How can we show that there exists infinitely many integer so that is a square?

- Sep 27th 2010, 01:37 AMOpalg
If , suppose for convenience that is a multiple of 3, Then , or If is large, then the left side will be close to and so must be close to . So go to the continued fraction calculator and plug in "3" in the square root box. You will then see a list of the convergents for the continued fraction expansion of , and you will notice that every fourth item in the list has an even numerator:

Let be the n'th term in that sequence. Then and the sequence grows by the inductive rules , . You should then be able to show by induction that . That gives an infinite family of integer solutions to the equation . Going back to the original problem, , so that gives infinitely many integers x such that is a square.